In this paper, motivated in part by the role of discrete groups of dilations in wavelet theory, we introduce and investigate the anisotropic Hardy spaces associated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky. Given a dilation $A$, that is an $ntimes n$ matrix all of whose eigenvalues $lambda$ satisfy $lambda>1$, define the radial maximal function $M^0_varphi f(x): = sup_{kinmathbb{Z}} (f*varphi_k)(x), qquadmathtext{where} varphi_k(x) = det A[UNK]^{-k} varphi(A^{-k}x).$ Here $varphi$ is any test function in the Schwartz class with $int varphi not=0$. For $0