One-dimensional adaptive Fourier decomposition, abbreviated as 1-D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher-order Szegö kernels of the context. In the present paper, we study multi-dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product-TM Systems; and the other uses Product-Szegö Dictionaries. With the Product-TM Systems approach, we prove that at each selection of a pair of parameters, the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product-Szegö dictionary approach, we show that pure greedy algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre-orthogonal Greedy Algorithm (P-OGA). We prove its convergence and convergence rate estimation, allowing a weak-type version of P-OGA as well. The convergence rate estimation of the proposed P-OGA evidences its advantage over orthogonal greedy algorithm (OGA). In the last part, we analyze P-OGA in depth and introduce the concept P-OGA-Induced Complete Dictionary, abbreviated as Complete Dictionary. We show that with the Complete Dictionary P-OGA is applicable to the Hardy H space on 2-torus. Copyright © 2016 John Wiley & Sons, Ltd.