The so-called structure-preserving methods which reproduce the fundamental properties like symplecticness, time reversibility, volume and energy preservation of the original model of the underlying physical problem became very important in recent years. It has been shown theoretically and experimentally, that these methods are superior to the standard integrators, especially in long term computation. In the paper the adaptivity issues are discussed for symplectic and reversible methods designed for integration of Hamiltonian systems. Molecular dynamics models and N-body problems, as Hamiltonian systems, are challenging mathematical models in many aspects; the wide range of time scales, very large number of differential equations, chaotic nature of trajectories, restriction to very small step sizes in time, etc. Recent results on variable step size integrators, multiple time stepping methods, regularization techniques with applications to classical and quantum molecular dynamics, to N- body atomic problems and planetary motion will be presented.