This dissertation begins with an introduction to finite-dimensional optimization and optimal control theory. It then proves lossless convexification for three problems: 1) a minimum time rendezvous using differential drag, 2) a maximum divert and landing, and 3) a general optimal control problem with linear state constraints and mixed convex and non-convex control constraints. Each is a unique contribution to the theory of lossless convexification. The first proves lossless convexification in the presence of singular controls and specifies a procedure for converting singular controls to the bang-bang type. The second is the first example of lossless convexification with state constraints. The third is the most general result to date. It says that lossless convexification holds when the state space is a strongly controllable subspace. This extends the controllability concepts used previously, and it recovers earlier results as a special case. Lastly, a few of the remaining research challenges are discussed. / text