Summary: New time-marching algorithms for solving initial value problems for nonlocal evolution equations are described. With respect to the space variable the discretization is based on a method of “approximate approximation” proposed by the second author. In time the algorithms are finite difference schemes of either the first or the second approximation order, whereas with respect to the space variables we use “approximate approximations” of an arbitrary high order. The algorithms are stable under mild restrictions to the time step which come from the nonlinear part of the equation. Some computational results and hints on crucial implementation issues are provided.