We establish the exponential convergence with respect to the L 1 -Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equationwhere (Z t ) t≥0 is a pure jump Lévy process whose Lévy measure ν fulfillsfor some constant κ 0 > 0, and the drift term b satisfies that for anywith some positive constants K 2 , l 0 and positive measurable function Φ 1 . The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process (X t ) t≥0 .where Φ 1 and Φ 2 are two nonnegative measurable functions, and l 0 ≥ 0 is a constant. For example, when Φ 2 (r) = K 2 r for some positive constantThis holds if the drift term b is dissipative outside some compact set. In particular, when Φ 1 (r) = K 1 r for some constant K 1 ≥ 0, it follows from (1.3) that for anywhich, along with (1.3), yields that the SDE (1.1) has a non-explosive and pathwise unique strong solution, see [1, Chapter 6, Theorem 6.2.3] (in the standard Lipschitz case) or [11, Theorem 2] and [25, Chapter 3, Theorem 115] (in the one-sided Lipschitz case). Note that, sincewe are sometimes concerned with only measurable drift term b, non-Lipschitz condition like B(K 1 , K 2 r, l 0 ) will also be adopted in our results below. The reader can refer to [8,20,22,26,32] and references therein for recent studies on the existence and uniqueness of strong solution to (1.1) with non-regular drift term. In particular, assuming that Z is the truncated symmetric α-stable process on R d with α ∈ (0, 2), and b is bounded and β-Hölder continuous with β > 1 − α/2, it was proved in [8, Corollary 1.4(i)] that the SDE (1.1) has a unique strong solution for each x ∈ R d . Furthermore, in the one-dimensional case, if α > 1, then the SDE (1.1) also enjoys a unique strong solution for each x ∈ R, even if the drift b is only bounded and measurable (see [26, Remark 1, p. 82]). Denote by ν the Lévy measure of the pure jump Lévy process Z. We assume that there is a constant κ 0 > 0 such that(1.4)Condition (1.4) was first used in [23] to study the coupling property of Lévy processes. It is satisfied by a large class of Lévy measures. For instance, iffor some z 0 ∈ R d and some ε > 0 such that ρ 0 (z) is positive and continuous on B(z 0 , ε), then such Lévy measure ν fulfills (1.4), see [24, Proposition 1.5] for details. Actually, as shown in Proposition 6.5, the condition (1.4) implies that there is a nonnegative measurable function ρ on R d such that ν(dz) ≥ ρ(z) dz andLet (P t ) t≥0 be the transition semigroup associated with the process (X t ) t≥0 . In this paper we are interested in the asymptotics of the Wasserstein-type distances (including the L 1 -Wasserstein distance and the total variation) between probability distributions δ x P t = P t (x, ·) and δ y P t = P t (y, ·) for any x, y ∈ R d , when the drift term b is dissipative outside some compact set, i.e. b satisfies B(Φ 1 (r), K 2 r, l 0 ) for some positive measurable function Φ 1 , and some const...