This dissertation focuses on the analysis of Stochastic Volatility (SV) models with leverage effect. We propose a general family of asymmetric SV (GASV) models and consider in detail two particular specifications within this family. The first one is the Threshold GASV (T-GASV) model which nests some of the most famous asymmetric SV models available in the literature with the errors being either Normal or GED.We also propose score driven GASV models with different assumptions about the error distribution, namely the Normal, Student-t or GED distributions, where the volatility is driven by the score of the lagged return distribution conditional on the volatility. Closed-form expressions of some statistical moments of interest of these two GASV models are derived and analyzed. We show that some of the parameters of these models cannot be properly identified by the moments usually considered when describing the stylized facts of financial returns, namely, excess kurtosis, autocorrelations of squares and cross-correlations between returns and future squared returns. As a byproduct, we obtain the statistical properties of those nested popular asymmetric SV models, some of which were previously unknown in the literature. By comparing the properties of these models, we are able to establish the advantages and limitations of each of them and give some guidelines about which model to implement in practice. We also propose the Stochastic News Impact Surface (SNIS) to represent the asymmetric response of volatility to positive and negative shocks in the context of SV models. The SNIS is useful to show the added flexibility of SV models over GARCH models when representing conditionally heteroscedastic time series with leverage effect. Analyzing the SNIS, we find that the asymmetric impact of the level disturbance on the volatility can be different depending on the volatility disturbance. Finally, we analyze the finite sample properties of a MCMC estimator of the parameters and volatilities of some restricted GASV models. Furthermore, estimating the restricted T-GASV model using this MCMC estimator, we show that one can correctly identify the true nested specifications which are popularly implemented in empirical applications. All the results are illustrated by Monte Carlo experiments and by fitting the models to both daily and weekly financial returns.