  A new class of explicit Euler schemes, which approximate stochastic
differential equations (SDEs) with superlinearly growing drift and diffusion
coefficients, is proposed in this article. It is shown, under very mild
conditions, that these explicit schemes converge in probability and in
$\mathcal{L}^p$ to the solution of the corresponding SDEs. Moreover, rate of
convergence estimates are provided for $\mathcal{L}^p$ and almost sure
convergence. In particular, the strong order $1/2$ is recovered in the case of
uniform $\mathcal{L}^p$-convergence.
