In structural dynamics, direct explicit and implicit integration algorithms are commonly used to solve the temporally discretized differential equations of motion for linear and nonlinear structures. The stability of different integration algorithms for linear elastic structures has been extensively studied for several decades. However, investigations of the stability applied to nonlinear structures are relatively limited and rather challenging. Recently, the authors proposed two systematic approaches using Lyapunov stability theory to investigate the stability property of direct integration algorithms of nonlinear dynamical systems. The first approach is a numerical one that transforms the stability analysis to a problem of convex optimization. The second approach investigates the Lyapunov stability of explicit algorithms considering the strictly positive real lemma. This paper reviews and compares these two Lyapunov-based approaches in terms of their merits and limitations.