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Let 
$(X,d,{\it\mu})$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions. Let 
$\mathscr{L}$ be a nonnegative self-adjoint operator on 
$L^{2}(X,d{\it\mu})$ satisfying a pointwise Gaussian upper bound estimate and Hölder continuity for its heat kernel. In this paper, we introduce the Hardy spaces 
$H_{\mathscr{L}}^{p}(X)$, 
$0<p\leq 1$, associated to 
$\mathscr{L}$ in terms of grand maximal functions and show that these spaces are equivalently characterised by radial and nontangential maximal functions.
