We apply several recently proposed tests for structural breaks in conditional variance and covariance dynamics. The tests apply to both the class of ARCH and SV type processes and allow for long memory features. We also apply them to data-driven volatility estimators using high-frequency data and suggest multivariate applications. In addition to testing for the presence of breaks, the statistics allow to identify the number of breaks and the location of multiple breaks. We study the size and power of the new tests under various realistic univariate and multivariate conditional variance models and sampling schemes. The paper concludes with an empirical analysis using data from the stock and FX markets for which we find multiple breaks associated with the Asian and Russian financial crises. We find changes in the dynamics and long memory of volatility in the samples prior and post the breaks.