Numerical Integrators for dynamical systems come in many forms: Euler, Runge-Kutta, Verlet, leapfrog, splitting schemes. Why are there so many, and which properties make one preferable to another? Accuracy per step is only part of the answer. For long simulations, what matters is whether the method preserves the structure of the underlying dynamics, such as conserved quantities, phase-space volume and time reversibility. We examine this using two systems: the Lotka-Volterra model and Molecular Dynamics.