These lectures introduce the modern theory and practical numerical methods for continuation of solutions of nonlinear problems depending upon parameters. Bifurcations are one of the many types of singularities that occur along such solution paths and their computation and methods for switching branches are treated. Homotopy methods and degree theory are introduced as are global Newton methods, constructive determination of Brouwer fixed points, periodic solutions of O.D.E.s, Hopf bifurcations, etc. The treatment is elementary and basic. Advanced calculus and linear algebra are sufficient to understand almost all of the material. A chapter showing applications and numerical methods for nonlinear O.D.E.s is included.