Backstepping is a mature and powerful Lyapunov-based design approach for a specific set of systems. Throughout the development over three decades, innovative theories and practices have extended backstepping to stabilization and tracking problems for nonlinear systems with growing complexity. The attractions of the backstepping-like approach are the recursive design processes and modularized design. A nonlinear system can be transferred into a group of simple problems and solved it by a sequential superposition of the corresponding approaches for each problem. To handle the complexities, backstepping designs always come up with adaptive control and robust control. The survey aims to review the milestone theoretical achievements among thousands of publications making the state-feedback backstepping designs of complex ODE systems to be systematic and modularized. Several selected elegant methods are reviewed, starting from the general designs, and then the finite-time control enhancing the convergence rate, the fuzzy logic system and neural network estimating the system unknowns, the Nussbaum function handling unknown control coefficients, barrier Lyapunov function solving state constraints, and the hyperbolic tangent function applying in robust designs. The associated assumptions and Lyapunov function candidates, inequalities, and the deduction key points are reviewed. The nonlinearity and complexities lay in state constraints, disturbance, input nonlinearities, time-delay effects, pure feedback systems, event-triggered systems, and stochastic systems. Instead of networked systems, the survey focuses on stand-alone systems.

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