Inspired by a question of Lov\'asz, we introduce a hierarchy of nested
semidefinite relaxations of the convex hull of real solutions to an arbitrary
polynomial ideal, called theta bodies of the ideal. For the stable set problem
in a graph, the first theta body in this hierarchy is exactly Lov\'asz's theta
body of the graph. We prove that theta bodies are, up to closure, a version of
Lasserre's relaxations for real solutions to ideals, and that they can be
computed explicitly using combinatorial moment matrices. Theta bodies provide a
new canonical set of semidefinite relaxations for the max cut problem. For
vanishing ideals of finite point sets, we give several equivalent
characterizations of when the first theta body equals the convex hull of the
points. We also determine the structure of the first theta body for all ideals.Comment: 26 pages, 3 figure