This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters p ∈ (0, ∞), q ∈ (0, ∞] and α ∈ R. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. For the Banach space regime p, q ≥ 1, we use this characterization to prove the existence of frames and Riesz sequences of dual molecules for the Triebel-Lizorkin spaces; the atoms are obtained by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of smoothness, decay and moment conditions.

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