We study Banach-space-valued martingale transforms and, in particular, characterize those Banach spaces for which the classical theorems of the real-valued case carry over. For example, if $B$ is a Banach space and $1 < p < \infty$, then there exists a positive real number $c_p$ such that $\|\epsilon_1d_1 + \cdots + \epsilon_n d_n \|_p \leq c_p \|d_1 + \cdots + d_n \|_p$ for all $B$-valued martingale difference sequences $d = (d_1, d_2,\cdots)$ and all numbers $\epsilon_1, \epsilon_2,\cdots$ in $\{-1, 1\}$ if and only if there is a symmetric biconvex function $\zeta$ on $B \times B$ satisfying $\zeta(0, 0) > 0$ and $\zeta(x, y) \leq | x + y | \text{if} | x | \leq 1 \leq | y |$.