We investigate innovative higher-order adaptive integrators for high-dimensional nonlinear evolution equations. First we discuss high-order split-step time integrators in a general Banach space framework and apply the results to prove convergence for the equations of motion associated with the multi-configuration time-dependent Hartree-Fock equations for the time-dependent Schroedinger equation. To improve the efficiency, we put forward pairs of embedded splitting formulae for error estimation and adaptive step-size selection. In the second part we investigate the properties of dissipative full discretizations for the equations of motion associated with models of flow and radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with specially adapted total-variation-diminishing Runge-Kutta time integrators with adaptive step-size control this yields reliable and efficient integrators for the underlying high-dimensional nonlinear evolution equations.