The asymptotic giant branch (AGB) evolution of stars is interesting from many points of view. It is the final evolutionary phase for the large majority of all stars in the Universe that have left the main sequence. During this stage they contribute to the chemical evolution of galaxies, contribute to the integrated starlight of many galaxies, may provide the only interstellar matter there is in some galaxies, and can, due to their high luminosities and ages, be used as important probes of galactic structure and dynamics. It is also the final evolutionary stage of our own star, the Sun, and hence touches on aspects beyond the immediate astronomical interest. Furthermore, these stars provide us with fascinating systems, where intricate interplays between physical and chemical processes take place, that we simply would like to understand. In this introductory outline our present knowledge of the AGB-star phenomenon is reviewed.

An overview of the structure and evolution of Asymptotic Giant Branch stars is presented. We focus upon the occurrence of thermal pulses, hot bottom burning and mass loss and summarize briefly the corresponding nucleosynthesis and mixing events. The interplay between these important evolutionary features is discussed.

We give a qualitative review of the nucleosynthesis occurring in AGB stars. We summarise some new calculations of intermediate mass stars which include all thermal pulses until the star is about to leave the AGB, as well as a detailed nucleosynthesis network. We will show that hot bottom burning delays, rather than prevents, the formation of carbon stars; those that form are not optically visible.

We investigate the influence of convective overshoot on stellar evolution models of the thermal pulse AGB phase with MZAMS = 3M⊙. An exponential diffusive overshoot algorithm is applied to all convective boundaries during all evolutionary stages.

We demonstrate that overshooting at the bottom of the pulse-driven convective zone, which forms in the intershell during the He-shell flash, leads to more efficient third dredge-up. Some overshoot at the bottom of the convective envelope removes the He-H discontinuity, which would otherwise prohibit the occurrence of the third dredge-up for this stellar mass. However, no correlation between the amount of envelope overshoot and the efficiency of the third dredge-up has been found.

Increasingly efficient third dredge-up eventually leads to a carbon star model. Due to the partial mixing efficiency in the overshoot region a 13C-pocket can form after the third dredge-up event which may be crucial for n-capture nucleosynthesis.

The third dredge-up phenomenon in asymptotic giant branch stars, responsible for the formation of C stars, is discussed based on detailed evolutionary model calculations. The structural readjustment of the star as dredge-up proceeds is analyzed, and its consequences on dredge-up predictions discussed. The question of how to obtain dredge-up in asymptotic giant branch models is also addressed. It is stressed, in particular, that the modeling of dredge-up requires some sort of extra-mixing to be applied below the envelope if the local Schwarzschild criterion is used to delimit convective zones.

We present recent improvements and results of an extensive analysis of the TP-AGB phase performed by means of a synthetic model (Marigo 1998a, b; Marigo et al. 1998a, b). The improvements concern: i) the use of a homogeneous and accurate set of analytical relations (Wagenhuber & Groenewegen 1998); ii) a new treatment of envelope burning in the most massive TP-AGB stars (M > 3.5M⊙) to account for the possible break-down of the core mass-luminosity relation; iii) a better treatment of the third dredge-up to infer if and when the process takes place.

Extensive calculations of synthetic TP-AGB models have been carried out over the mass range (0.8M⊙ ÷ 5M⊙) and for three sets of initial metallicity (Z = 0.019, Z = 0.008, Z = 0.004). The formation of carbon stars is investigated addressing the following issues: a) the reproduction of the observed luminosity functions of carbons stars in both Magellanic Clouds, and b) the formation of very bright and optically obscured carbon stars.

Primitive meteorites contain dust grains that predate the Solar System, formed in stellar atmospheres and thus represent samples of ancient Stardust. Among the presolar grain types identified so far, corundum (Al2O3) and silicon carbide (SiC) are inferred to originate from AGB stars. Corundum grains carry the signatures of core H burning in their O isotopes and of shell H burning during the AGB phase in the form of extinct 26Al. In presolar SiC, most of which originated from carbon stars, the C and N isotopes and 26Al reflect core and shell H burning and shell He burning. In addition, many elements that carry the isotopic signature of neutron capture have also been measured. Most individual grains show excesses in 29Si and 30Si, but the contribution from neutron capture is only a minor effect and the major effect is due to galactic heterogeneity. Noble gases and the elements Ba, Nd, Sm, and Dy are measured in ”bulk samples”, collections of many grains. Their measured isotopic patterns are well reproduced by models of the s-process in AGB stars. Recently, the isotopic analysis of Sr, Zr and Mo in single SiC grains has been made possible by resonance ionization mass spectrometry. These measurements also point to low-mass AGB stars as the most likely sources. Specifically, large 96Zr depletions in some grains indicate that the 22Ne(α, n) source was not active in the grains' parent stars.

The presence of excess 12C, 13C, 14N, Li, and the heavy s-process elements in the photospheres of AGB stars betray the operation of Eland 4He-burning, hot-bottom burning, and slow neutron-capture nucleosynthesis. Careful abundance analyses of these various elements provides insight into both nucleosynthesis events and mixing processes which occur deep within the AGB star. Recent results for the s-process are discussed.

I review the current status of cool O-rich giant star spectral modelling, stressing some specific problems like line profiles and molecular data completeness. I discuss recent progress in molecular line lists (TiO, VO) and describe recent attempts in modelling normal M-giants as well as AGB star spectra, using static and hydrodynamic models. I anticipate further advances in the near future, with better radiative transfer in hydrodynamic models, that will allow detailed comparisons of models and observations, opening the possibility of deriving stellar parameters from spectroscopic and interferometric observations.

We have completed a grid of spherically symmetric AGB star atmospheres using the state of the art spectral synthesis code PHOENIX. Models are constructed for stars with masses of 1 M⊙ and 1.5 M⊙, spanning the range 10 to 3300 L⊙ in luminosity and 2500 to 5200 K in effective temperature. We find that grains of Al2O3 and CaTiO3 among other species form in atmospheres cooler than Teff = 3000 K. In the coolest models the grains cause a weakening of the TiO absorption features in the red and near infrared of up to 30% through both a depression of the continuum and a depletion of the TiO number abundance. We use spectrophotometric observations from a number of catalogs to determine effective temperature – spectral class and effective temperature – color relationships. We also compare synthetic colors calculated from our models with observations of M giants on Wing's 8-color narrow-band system of classification photometry.

We present the first results of a wide observational program intended to study the effects of hot-bottom burning and of the third dredge-up in the chemical evolution of the most luminous AGB stars in our Galaxy. The main goal is to provide new observational constraints to the models through the analysis of lithium and s-process element abundances in a carefully selected sample of galactic AGB stars. Our results indicate that, unlike observed in the Magellanic Clouds, the most massive AGB candidates in our Galaxy do not show any significant enrichment in s-process elements. In addition, only some of them are found to be Li-rich.

In the last 3 decades, infrared surveys have discovered new classes of AGB stars, and significantly improved our knowledge of their statistical and physical properties, often in conjunction with surveys in other spectral ranges (mainly OH maser lines). After a short description of the major past and present infrared surveys and their most commonly used analysis tools, their impact is illustrated by a few examples such as the characterisation of the populations of the Bulge and Magellanic Clouds and the distribution of carbon-rich stars in the galactic disc. In conclusion, further all-sky surveys in the still unexplored spectral regions between 2 and 10 μm are strongly advocated as well as the development of ground-based telescopes dedicated to deeper surveys and monitoring of the variability.

The most direct evidence for the pulsation mode of Mira and SR variables comes from the linear diameters derived from individual parallaxes and angular diameters, together with the observed periods. The data now available strongly suggest that most, if not all, Miras and SRs pulsate in the same mode. If comparison is made with conventional models, this mode is the first overtone. Uncertainties in deriving the pulsational radius of a Mira from the observations do not appear significant enough to affect this conclusion. Evidence from SRs and Miras in globular clusters supports this conclusion and shows that the pulsation mode does not depend on metallicity. Analysis of light curves leads to broadly the same conclusions, if less directly. The light curves indicate that periodicities other than the main one may also be present and that mode switching may occur in some stars.

The HIPPARCOS satellite provides astronomical data for about one thousand Long Period Variable stars (LPVs) from which kinematics properties and luminosity calibrations in several bandpasses are deduced using an appropriate method. Several results are deduced: a classification of the LPVs and its relation with the classification from the light curves, a calibration of luminosities inducing properties (age, mass, etc.) along the AGB and a comparison of oxygen to carbon-rich stars.

Three issues, which are to my opinion the main obstacles in modeling the variability of Miras and Long Period Variables (LPVs), are presented and discussed:

1. a. The initial static models - Possible influences of the evolutionary details, the helium shell flashes and the use of the core-mass luminosity relation.

2. b. Modeling the convective energy transport - The mixing-length theory, its limitations and validity. First attempts for two-dimensional calculations of the convective flow in AGB stars.

3. c. The surface conditions - Introduction to the main difficulties and uncertainties. What should be done in the future.

Finally, the ancient problem, concerning the mode of pulsation, in Miras, is discussed using the recent radii observations confronted with the results of our full non-linear calculations.

We present a detailed period analysis for 98 red semiregular variables by means of Fourier and wavelet analysis of long-term visual observations carried out by amateur astronomers. The overwhelming majority of the studied stars show multiperiodic behaviour. We found two significant periods in 62 variables, while there are definite signs of three periods in 13 stars. 20 stars turned out to be monoperiodic with small instabilities in the period. Since this study deals with the general trends, we want to find only the most dominant periods.

The distribution of periods and period ratios is examined in the (log P1, log P0/P1) plots. Three significant and two less obvious sequences are present which can be explained as the straight consequence of different pulsational modes. This hypothesis is supported by the multiperiodic variables with three periods. A clear distinction between C-rich and O-rich stars has been found in these diagrams suggesting a connection between the chemistry and pulsational characteristics.

I describe here results from high-angular resolution imaging studies of o Ceti (Mira). In 1983, we discovered that the atmosphere of the prototype of Mira-type variables is not symmetric. Since then, a number of multiwavelength high-angular resolution observations have confirmed the presence of asymmetries in Mira's atmosphere, and detected asymmetries in the atmospheres of other Mira-type variables. The high-angular resolution images of Mira obtained over the past fifteen years, including recent HST observations, show that the strength and shape of the asymmetries change as a function of wavelength and time. Plausible mechanisms for these asymmetries include hot spots, nonspherical pulsations, interaction with the companion and bipolar outflow. The presence of asymmetries in Miras could have serious impact on evolutionary models, and on the development of model atmospheres.