Stochastic volatility (SV) models provide a means of tracking and forecasting the variance of financial asset returns. While SV models have a number of theoretical advantages over competing variance modelling procedures they are notoriously difficult to estimate. The distinguishing feature of the SV estimation literature is that those algorithms that provide accurate parameter estimates are conceptually demanding and require a significant amount of computational resources to implement. Furthermore, although a significant number of distinct SV specifications exist, little attention has been paid to how one would choose the appropriate specification for a given data series. Motivated by these facts, a likelihood based joint estimation and specification testing procedure for SV models is introduced that significantly overcomes the operational issues surrounding existing estimators. The estimation and specification testing procedures in this thesis are made possible by the introduction of a discrete nonlinear filtering (DNF) algorithm. This procedure uses the nonlinear filtering set of equations to provide maximum likelihood estimates for the general class of nonlinear latent variable problems which includes the SV model class. The DNF algorithm provides a fast and accurate implementation of the nonlinear filtering equations by treating the continuously valued state-variable as if it were a discrete Markov variable with a large number of states. When the DNF procedure is applied to the standard SV model, very accurate parameter estimates are obtained. Since the accuracy of the DNF is comparable to other procedures, its advantages are seen as ease and speed of implementation and the provision of online filtering (prediction) of variance. Additionally, the DNF procedure is very flexible and can be used for any dynamic latent variable problem with closed form likelihood and transition functions. Likelihood based specification testing for non-nested SV specifications is undertaken by formulating and estimating an encompassing model that nests two competing SV models. Likelihood ratio statistics are then used to make judgements regarding the optimal SV specification. The proposed framework is applied to SV models that incorporate either extreme returns or asymmetries.