Incompatible elasticity is a term coined in the 1950s to describe the continuum theory of elastic solids in the presence of defects; the point of view, which was novel at that time, is that defects modify the intrinsic geometry of the body, making it incompatible with the ambient space. Incompatible elasticity has seen a renewed interest in recent years in the context of both natural and human-made systems, which can be viewed as metrically frustrated. From a mathematical point of view, incompatible elasticity can be described as a problem of optimally embedding one Riemannian manifold (the body) into another Riemannian manifold (the ambient space). In this lecture I will present the mathematical formulation of incompatible elasticity, review some recent results, such incompatible plate and shell theories and the homogenization of defects, and mention some open problems.