Cone manifolds are defined and several standard geometric techniques for Riemannian manifolds are generalized to this setting. Smoothing techniques for approximating cone manifolds by smooth Riemannian manifolds with bounded sectional curvature are discussed. This involves some quite explicit curvature computations. The connection is then made between branched covers and cone manifolds by showing that cone manifold structures lift to a branched cover. Topological results concerning existence of incompressible tori and Seifert-fibered spaces in branched covers are then obtained by lifting cone manifold structures to a branched cover, smoothing the cone manifold structure to a bounded curvature metric, then using differential-geometric techniques on the smooth manifold. These results are then used in several explicit examples.