I will discuss the characterization of convex sets in m which can be represented by Linear Matrix Inequalities, i.e., as feasible sets of semidefinite programmes. There is a simple necessary condition, called rigid convexity, which has been shown to be sufficient for sets in the plane and is conjectured to be sufficient (in a somewhat weakened sense) for any m. This should be contrasted with the situation for matrix convex sets that will feature in the talk of Scott McCullough, where all the available evidence suggests that any matrix convex set with noncommutative algebraic boundary admits an LMI representation. This is a joint work with Bill Helton.