Consider an elastic curve moving in three-dimensional space, driven by geometric forces and subject to a geometric constraint. Its potential energy is given by the square integral of the curvature, and the constraint is that the curve is inextensible. The equations of motion form a system of three nonlinear Schrödinger equations for the configuration and momentum of the curve, coupled to an elliptic boundary value problem for its tension. The system has interesting special solutions, including large families of knotted equlibria and solitary waves. While slow-moving solitary waves are unstable, they stabilize at higher wave speed; I will try to explain the mechanism. Less is known about general solutions, but I will describe a result on existence and uniqueness of solutions for sufficiently smooth initial data. Time permitting, I will discuss some open questions, including a conjecture for the geometric Schrödinger operator that appears in the tension equation.