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This paper studies the problem of stabilizing reference trajectories (also called as the trajectory tracking problem) for underactuated marine vehicles under predefined tracking error constraints. The boundary functions of the predefined constraints are asymmetric and time-varying. The time-varying boundary functions allow us to quantify prescribed performance of tracking errors on both transient and steady-state stages. To overcome difficulties raised by underactuation and nonzero off-diagonal terms in the system matrices, we develop a novel transverse function control approach to introduce an additional control input in backstepping procedure. This approach provides practical stabilization of any smooth reference trajectory, whether this trajectory is feasible or not. By practical stabilization, we mean that the tracking errors of vehicle position and orientation converge to a small neighborhood of zero. With the introduction of an error transformation function, we construct an inverse-hyperbolic-tangent-like barrier Lyapunov function to show practical stability of the closed-loop systems with prescribed transient and steady-state performances. To deal with unmodeled dynamic uncertainties and external disturbances, we employ neural network (NN) approximators to estimate uncertain dynamics and present disturbance observers to estimate unknown disturbances. Subsequently, we develop adaptive control, based on NN approximators and disturbance estimates, that guarantees the prescribed performance of tracking errors during the transient stage of on-line NN weight adaptations and disturbance estimates. Simulation results show the performance of the proposed tracking control.