This paper investigates a variety of features exhibited by the amplitude of stock returns. Some of these "stylized facts" have already attracted a great deal of attention from researchers, while some others have been documented only recently. - Horizontal dependence of volatility: Volatility is not constant, it is mean-reverting and it tends to cluster (meaning that high volatility is likely to be followed by high volatility periods, and vice-versa). Moreover, volatility exhibits a significant autocorrelation even for very long lags; it is therefore said to have long-memory. - Extreme events: The number of large (either positive or negative) returns is far bigger than what is expected on the basis of modern finance theory - stated differently, the returns distribution has fat tails. Moreover, shocks have a strong impact on volatility and generally lead to a number of aftershocks - this is again a feature that most financial models are unable to replicate. - The "leverage effect" (also known as the asymmetric volatility phenomenon): Volatility has a negative correlation with returns. Moreover, this relation is asymmetric: when returns are negative, volatility increases rapidly; but, when returns are positive, volatility decreases to a much lesser extent. - Vertical dependence and asymmetric vertical dependence: Volatility measured at different frequencies (e.g. intraday, daily or monthly) has different information content. Low-frequency volatility has a stronger explanatory power on subsequent high-frequency volatility than the other way around. Furthermore, an increase in volatility at low-frequencies has more impact on volatility at high-frequencies than a decrease. This paper extends the existing literature by studying all these stylized facts for both emerging and mature markets, and for both bull and bear market conditions. We are able to confirm the existence of all those stylized facts and to describe them in a more precise way than in the existing literature.