We investigate power graphs of quasigroups. The power graph of a quasigroup takes the elements of the quasigroup as its vertices, and there is an edge from one element to a second distinct element when the second is a left power of the first. We first compute the power graphs of small quasigroups (up to four elements). Next we describe quasigroups whose power graphs are directed paths, directed cycles, in-stars, out-stars, and empty. We do so by specifying partial Cayley tables, which cannot always be completed in small examples. We then consider sinks in the power graph of a quasigroup, as subquasigroups give rise to sinks. We show that certain structures cannot occur as sinks in the power graph of a quasigroup. More generally, we show that certain highly connected substructures must have edges leading out of the substructure. We briefly comment on power graphs of Bol loops.