  We survey recent developments in the field of complexity of pathwise
approximation in $p$-th mean of the solution of a stochastic differential
equation at the final time based on finitely many evaluations of the driving
Brownian motion. First, we briefly review the case of equations with globally
Lipschitz continuous coefficients, for which an error rate of at least $1/2$ in
terms of the number of evaluations of the driving Brownian motion is always
guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate
that giving up the global Lipschitz continuity of the coefficients may lead to
a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an
arbitrary slow decay of the smallest possible error that can be achieved on the
basis of finitely many evaluations of the driving Brownian motion. Finally, we
turn to recent positive results for equations with a drift coefficient that is
not globally Lipschitz continuous. Here we focus on scalar equations with a
Lipschitz continuous diffusion coefficient and a drift coefficient that
satisfies piecewise smoothness assumptions or has fractional Sobolev regularity
and we present corresponding complexity results.
