  In this article, we consider the problem of sampling from a probability
measure $\pi$ having a density on $\mathbb{R}^d$ known up to a normalizing
constant, $x\mapsto \mathrm{e}^{-U(x)} / \int_{\mathbb{R}^d} \mathrm{e}^{-U(y)}
\mathrm{d} y$. The Euler discretization of the Langevin stochastic differential
equation (SDE) is known to be unstable in a precise sense, when the potential
$U$ is superlinear, i.e. $\liminf_{\Vert x \Vert\to+\infty} \Vert \nabla U(x)
\Vert / \Vert x \Vert = +\infty$. Based on previous works on the taming of
superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted
Langevin Algorithm (TULA) and obtain non-asymptotic bounds in $V$-total
variation norm and Wasserstein distance of order $2$ between the iterates of
TULA and $\pi$, as well as weak error bounds. Numerical experiments are
presented which support our findings.
