This article contributes to the embryonic literature on the relations between Bitcoin and conventional investments by studying return and volatility spillovers between this largest cryptocurrency and four asset classes (equities, stocks, commodities, currencies and bonds) in bear and bull market conditions. We conducted empirical analyses based on a smooth transition VAR GARCH-in-mean model covering daily data from 19 July 2010 to 31 October 2017. We found significant evidence that Bitcoin returns are related quite closely to those of most of the other assets studies, particularly commodities, and therefore, the Bitcoin market is not isolated completely. The significance and sign of the spillovers exhibited some differences in the two market conditions and in the direction of the spillovers, with greater evidence that Bitcoin receives more volatility than it transmits. Our findings have implications for investors and fund managers who are considering Bitcoin as part of their investment strategies and for policymakers concerned about the vulnerability that Bitcoin represents to the stability of the global financial system.