This paper aims to assess dynamic tail risk exposure in the hedge fund sector using daily data. We use a copula function to model both lower and upper tail dependence between hedge-fund and broad-market returns as a function of market uncertainty. We proxy the latter by means of a single index that combines the options-implied market volatility, the volatility risk premium, and the swap and term spreads. We find substantial time-variation in both lower- and upper-tail dependence, even for hedge fund styles that exhibit little unconditional tail dependence. In particular, dependence between hedge fund and equity market returns decreases in both tails significantly with market uncertainty. There are only a few styles that feature neither unconditional nor conditional tail dependence, e.g., convertible arbitrage and equity market neutral. We also fail to observe any tail dependence with bond and currency markets, though we find strong evidence that the lower-tail risk exposure of macro hedge funds to commodity markets increases with liquidity risk. Finally, further analysis shows mixed evidence on how much hedge funds contribute to systemic risk.