In this Dissertation, computational and analytic methods are presented to address nonlinear systems with applications in structural and celestial mechanics. Scalar Homotopy Methods (SHM) are first introduced for the solution of general systems of nonlinear algebraic equations. The methods are applied to the solution of postbuckling and limit load problems of solids and structures as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. In many problems, instead of simply adopting a root solving method, it is useful to study the particular problem in more detail in order to establish an especially efficient and robust method. Such a problem arises in satellite geodesy coordinate transformation where a new highly efficient solution, providing global accuracy with a non-iterative sequence of calculations, is developed. Simulation results are presented to compare the solution accuracy and algorithm performance for applications spanning the LEOtoGEO range of missions. Analytic methods are introduced to address problems in structural mechanics and astrodynamics. Analytic transfer functions are developed to address the frequency domain control problem of flexible rotating aerospace structures. The transfer functions are used to design a Lyapunov stable controller that drives the spacecraft to a target position while suppressing vibrations in the flexible appendages. In astrodynamics, a Taylor series based analytic continuation technique is developed to address the classical two-body problem. A key algorithmic innovation for the trajectory propagation is that the classical averaged approximation strategy is replaced with a rigorous series based solution for exactly computing the acceleration derivatives. Evidence is provided to demonstrate that high precision solutions are easily obtained with the analytic continuation approach. For general nonlinear initial value problems (IVPs), the method of Radial Basis Functions time domain collocation (RBF-Coll ) is used to address strongly nonlinear dynamical systems and to analyze short as well as long-term responses. The algorithm is compared against, the second order central difference, the classical Runge-Kutta, the adaptive Runge-Kutta-Fehlberg, the Newmark-?, the Hilber-Hughes-Taylor and the modified Chebyshev-Picard iteration methods in terms of accuracy and computational cost for three types of problems; (1) the unforced highly nonlinear Duffing oscillator, (2) the Duffing oscillator with impact loading and (3) a nonlinear three degrees of freedom (3-DOF) dynamical system. The RBF-Collmethod is further extended for time domain inverse problems addressing fixed time optimal control problems and Lamberts orbital transfer problem. It is shown that this method is very simple, efficient and very accurate in obtaining the solutions. The proposed algorithm is advantageous and has promising applications in solving general nonlinear dynamical systems, optimal control problems and high accuracy orbit propagation in celestial mechanics.