Continuum robots, as highly coupled nonlinear multivariable systems, suffer from significant model inaccuracies that limit precise motion control, especially in surgical and inspection tasks. To address this, we propose a neural network-based adaptive trajectory tracking framework. Radial basis function (RBF) neural networks are used to approximate dynamic uncertainties—gravity, elasticity, Coriolis forces, and external disturbances—while a norm lower bound for the inertia matrix is established. These are integrated into an adaptive backstepping controller that ensures globally uniformly ultimately bounded stability. Simulation results confirm the method’s excellent tracking accuracy and strong robustness against unmodeled dynamics and disturbances.