Let $M_k(0, T)$ denote the $k$th (factorial) moment of the number of zero crossings in time $T$ by a stationary Gaussian process. We present a necessary and sufficient condition for $M_k(0, T)$ to be finite. This condition is then applied to processes whose covariance functions $\rho(t)$ satisfy the local condition. $$\rho(t) = 1 - \frac{t^2}{2} + \frac{C|t|^3}{6} + o|t|^3$$ for $t$ near zero $(C > 0)$. In this case we show all the crossing moments $M_k(0, T)$ are finite. In the course of the proof of this result, we point out an error which vitiates the related work of Piterbarg (1968) and Mirosin (1971, 1973, 1974a, 1974b). We also find a counterexample to Piterbarg's results.