Let $(\Omega, F, P)$ be a probability space, $N^2 = N \times N$ denote the set of parameters with the partial order defined by $(m_1, n_1)\leq (m_2, n_2)$ if and only if $m_1\leq m_2$ and $n_1\leq n_2$ $(m_1, n_1, m_2, n_2\in N)$. Let $F_{mn}$ be an increasing family of sub-$\delta$-fields of $F$ satisfying the usual condition $(F_4)$ and $(M_{mn}, F_{mn})$ a two-parameter martingale taking values in a Banach space $(B, \| \cdot \|)$. In this paper we investigate the interrelation between geometric properties of Banach spaces and Martingale convergence theorems. Moreover we also study Marcinkiewicz-Zygmund's type strong law of large numbers for two-parameter Banach-valued martingales and the integrability of two-parameter Banach-valued martingale maximal functions.