Summary: The problem of finding the unique frequency and shape, including the core, of spirals in excitable media is studied. A leading-order approximation of the problem is formulated as a nonlinear eigenvalue problem involving only ordinary differential equations. Shooting and comparison arguments are used to show why solutions of the leading-order approximate problem exist, and some of their quantitative features are explored numerically. Because of inaccuracies of the results in certain parameter ranges, an improved formulation of the problem is proposed. When solved numerically, as in the special case of symmetric spirals, solutions of the improved formulation differ significantly from solutions of the leading-order problem, but agree closely with the numerical solution of the full partial differential equation system.