In this chapter two robust control strategies for nonlinear dynamic systems are presented. This chapter begins with a brief review of the most common kind of uncertainties found nowadays by considering their mathematical and topological foundations. The kinds of uncertainties explained in this chapter are basically parametric, nonparametric, matched and mismatched uncertainties describing the mathematical properties, in specific, the topological properties taking into consideration that recently there are a plenty of mathematical representations due to the vast amount of uncertainties found in nonlinear and other kinds of complex dynamical systems. One important result explained in this chapter is that the well-known robust control Lyapunov functions (RCLFs) are explained by taking into consideration that the closed loop robust stability and the robust control law are found by an appropriate construction of the RCLF. Besides in this chapter is explained the theoretical background of the design of robust controllers by considering uncertainties in nonlinear dynamic systems, and by implementing the respective Lyapunov functional a suitable robust controller can be obtained in order to ensure robust stability of a closed loop system. Besides, the design of optimal robust controllers is explained by taking into consideration that a performance functional along with the respective Hamiltonian is shown in order to design the robust optimal control law by taking into consideration the uncertainties in the system. One of the main contributions of this chapter is that a robust controller is designed for systems in the Euler-Lagrange formulation by designing the Lagrangian in order to obtain the required formulation. The mass and inertia uncertainties are considered in this study with the respective topological definition, and by using an appropriate Lyapunov functional, the robust control law is found. This control law can be implemented in a variety of mechanical and robotic systems for trajectory tracking purposes. Finally, an optimal robust controller for mechanical systems is presented in this chapter by taking into consideration a Lyapunov functional and a performance functional for the robust optimal control law design. This robust optimal controller is implemented in a nonlinear forced mass-spring system, corroborating a high performance.