Voters have strong incentives to increase their influence by trading votes, acquiring others' votes when preferences are strong in exchange for giving votes away when preferences are weak. But is vote trading welfare-improving or welfare-decreasing? For a practice long believed to be central to collective decisions, the lack of a clear answer is surprising. We review the theoretical literature and, when available, its related experimental tests. We begin with the analysis of logrolling - the exchange of votes for votes. We then focus on vote markets, where votes can be traded against a numeraire. We conclude with procedures allowing voters to shift votes across decisions - to trade votes with oneself only. We find that vote trading and vote markets are typically inefficient; more encouraging results are obtained by allowing voters to allocate votes across decisions.