This paper introduces a new class of multivariate volatility models that utilizes high-frequency data. We discuss the models' dynamics and highlight their differences from multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models. We also discuss their covariance targeting specification and provide closed-form formulas for multi-step forecasts. Estimation and inference strategies are outlined. Empirical results suggest that the HEAVY model outperforms the multivariate GARCH model out-of-sample, with the gains being particularly significant at short forecast horizons. Forecast gains are obtained for both forecast variances and correlations.