This paper is concerned with a class of nonlinear stochastic wave
equations in $\mathbb{R}^d$ with $d \leq 3$, for which the nonlinear terms are
polynomial of degree $m$. As an example of the nonexistence of a global
solution in general, it is shown that there exists an explosive solution of
some cubically nonlinear wave equation with a noise term. Then the existence
and uniqueness theorems for local and global solutions in Sobolev space $H_1$
are proven with the degree of polynomial $m \leq 3$ for $d = 3$, and $m \geq 2$
for $d = 1$ or 2.