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We prove a general criterion providing sufficient conditions under which atime-discretiziation of a given Stochastic Differential Equation (SDE) is auniform in time approximation of the SDE. The criterion is also, to a certainextent, discussed in the paper, necessary. Using such a criterion we thenanalyse the convergence properties of numerical methods for solutions of SDEs;we consider Explicit and Implicit Euler, split-step and (truncated) tamed Eulermethods. In particular, we show that, under mild conditions on the coefficientsof the SDE (locally Lipschitz and strictly monotonic), these methods produceapproximations of the law of the solution of the SDE that converge uniformly intime. The theoretical results are verified by numerical examples.

Uniform in time convergence of numerical schemes for stochastic  differential equations via Strong Exponential stability: Euler methods,  Split-Step and Tamed Schemes