We propose flexible models for multivariate realized volatility dynamics which involve generalizations of the Box-Cox transform to the matrix case. The matrix Box-Cox model of realized covariances (MBC-RCov) is based on transformations of the covariance matrix eigenvalues, while for the Box-Cox dynamic correlation (BC-DC) specification the variances are transformed individually and modeled jointly with the correlations. We estimate transformation parameters by a new multivariate semiparametric estimator and discuss bias-corrected point and density forecasting by simulation. The methods are applied to stock market data where excellent in-sample and out-of-sample performance is found.