A systematic study of geodesics in spacetimes containing cosmic strings is presented. Different models of spacetime of a single string as well as models of asymptotically conical spacetimes describing isolated matter systems pierced by cosmic strings are studied. Geodesics in the flat conical, conical Schwarzschild and Lense-Thirring spacetimes are described in detail. The problem of two-body motion in the presence of a cosmic string is also discussed. Many new kinematical effects arising from the conical structure of spacetime are found.