We review some empirical facts of financial markets that have motivated the rough volatility paradigm for modelling financial volatility, both from the point of view of financial time series and options pricing. Motivated by empirical evidence from the joint behavior of realized volatility time series, we propose to model the joint dynamics of log-volatilities using a multivariate fractional Ornstein-Uhlenbeck process. This model is a multivariate version of the Rough Fractional Stochastic Volatility model proposed in Gatheral, Jaisson, and Rosenbaum, Quant. Finance, 2018. It allows for different Hurst exponents in the different marginal components and non trivial interdependencies. We discuss the main features of the model, propose parameter estimators, derive their asymptotic theory and perform a simulation study that confirms the asymptotic theory in finite sample. We carry out an extensive empirical investigation on empirical realized volatility time series, showing that these time series are strongly correlated and can exhibit asymmetries in their empirical cross-covariance function, accurately captured by our model. These asymmetries lead to spillover effects, which we derive analytically within our model and compute based on empirical estimates of model parameters. Moreover, in accordance with the existing literature, we observe behaviors close to non-stationarity and rough trajectories.