Control algorithms suitable for online implementation in engineering applications, such as aerospace and mechanical vehicles, often require adherence to physical state and control constraints. Additionally, the chosen algorithms must provide robustness to uncertainty affecting both the system dynamics and the constraints. As further autonomy is built into these systems, the algorithms must be capable of blending multiple operational modes without violating the intrinsic constraints. Further, for real-time applications, the implemented control algorithms must be computationally efficient and reliable. The research in this thesis approaches these application needs by building upon the framework of MPC (Model Predictive Control). The MPC algorithm makes use of a nominal dynamics model to predict and optimize the response of a system under the application of a feedforward control policy, which is computed online in a finite-horizon optimization problem. The MPC algorithm is quite general and can be applied to linear and nonlinear systems and include explicit state and control constraints. The finite-horizon optimization is advantageous given the finite online computational capabilities in practical applications. Further, recursively re-solving the finite-horizon optimization in a compressing- or receding-horizon manner provides a form of closed-loop control that updates the feedforward control policy by setting the nominal state at re-solve to the current actual state. However, uncertainty between the nominal model and the actual system dynamics, along with constraint uncertainty can cause feasibility, and hence, robustness issues with the traditional MPC algorithm. In this thesis, an R-MPC (Robust and re-solvable MPC) algorithm is developed for uncertain nonlinear systems to address uncertainty affecting the dynamics. The R-MPC control policy consists of two components: the feedforward component that is solved online as in traditional MPC; and a separate feedback component that is determined offline, based on a characterization of the uncertainty between the nominal model and actual system. The addition of the feedback policy generates an invariant tube that ensures the actual system trajectories remain in the proximity of the nominal feedforward trajectory for all time. Further, this tube provides a means to theoretically guarantee continued feasibility and thus re-solvability of the R-MPC algorithm, both of which are required to guarantee asymptotic stability. To address uncertainty affecting the state constraints, an SR-MPC (Safety-mode augmented R-MPC) algorithm is developed that blends a reactive safety mode with the R-MPC algorithm for uncertain nonlinear systems. The SR-MPC algorithm has two separate operational modes: standard mode implements a modified version of the R-MPC algorithm to ensure asymptotic convergence to the origin; safety mode, if activated, guarantees containment within an invariant set about a safety reference for all time. The standard mode modifies the R-MPC algorithm with a special constraint to ensure safety-mode availability at any time. The safety-mode control is provided by an offline designed control policy that can be activated at any time during standard mode. The separate, reactive safety mode provides robustness to unexpected state-constraint changes; e.g., other vehicles crossing/stopping in the feasible path, or unexpected ground proximity in landing scenarios. Explicit design methods are provided for implementation of the R-MPC and SR-MPC algorithms on a class of systems with uncertain nonlinear terms that have norm-bounded derivatives. Further, a discrete SR-MPC algorithm is developed that is more broadly applicable to real engineering systems. The discrete algorithm is formulated as a second-order cone program that can be solved online in a computationally efficient manner by using interior-point algorithms, which provide convergence guarantees in finite time to a prescribed level of accuracy. This discrete SR-MPC algorithm is demonstrated in simulation of a spacecraft descent toward a small asteroid where there is an uncertain gravity model, as well as errors in the expected surface altitude. Further, realistic effects such as control-input uncertainty, sensor noise, and unknown disturbances are included to further demonstrate the applicability of the discrete SR-MPC algorithm in a realistic implementation.