We propose several connectedness measures built from pieces of variance decompositions, and we argue that they provide natural and insightful measures of connectedness among financial asset returns and volatilities. We also show that variance decompositions define weighted, directed networks, so that our connectedness measures are intimately-related to key measures of connectedness used in the network literature. Building on these insights, we track both average and daily time-varying connectedness of major U.S. financial institutions' stock return volatilities in recent years, including during the financial crisis of 2007-2008.