The convergence properties of a class of high-order semi-Lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions. We review the general convergence results for discrete-time approximation schemes belonging to that class and we prove some a priori estimates in $L^\infty$ and $L^2$ for the rate of convergence of fully discrete schemes. We prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions. Several examples of schemes and numerical tests are presented.