In this dissertation we present extensions of Rice's formula for the expected zero-crossing rate of a Gaussian process to some useful non-Gaussian cases. In particular, we extend Rice's formula to the class of stationary processes which are a monotone transformation of a Gaussian process, to countable mixtures of Gaussians, and to products of independent Gaussian processes. In all the above mentioned cases the expected zero-crossing rates are given for both continuous time and discrete time processes. We also investigate the application of parametric filtering, using zero-crossing count statistics, to the problem of frequency estimation in a mixed spectrum model and the application of mean- level-crossing counts of the envelope of a Gaussian process to a radar detection problem. For the radar problem we prove asymptotic normality of the level-crossings of the envelope of a Gaussian process and provide and expression for the asymptotic variance.