The authors present a general solution of the equations of motion for a classical relativistic string in Minkowski space. For an 'open' string the solution is determined by a single real null curve which satisfies a quasiperiodicity condition. A formula is deduced which generates all such curves in terms of freely specifiable functions. For a 'closed' string two such curves are required. Their theory establishes a rigorous basis for the numerical investigation of 'cosmic strings', and also forms a starting point for a constraint-free Lorentz covariant analysis of the quantum relativistic theory.