We discuss two more universal features of stock markets: the so-called leverage effect (a negative correlation between past returns and future volatility), and the increased downside correlations. For individual stocks, the leverage correlation can be rationalized in terms of a new ‘retarded’ model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific market panic phenomenon seems to be necessary to account for the observed amplitude of the effect. As for the increase of correlations in highly volatile periods, we investigate how much of this effect can be explained within a simple non-Gaussian one-factor description with time independent correlations. In particular, this one-factor model can explain the level and asymmetry of empirical exceedance correlations, which reflects the fat-tailed and negatively skewed distribution of market returns.