In this talk, we provide explicit, non-asymptotic perturbation bounds for Metropolis-Hastings algorithms when the likelihood in the acceptance probability is subject to an approximation error. This type of approximation error may arise when the likelihood of the target density is intractable. A prominent example is the Monte Carlo within Metropolis algorithm (MCWM) for latent variable models. We focus on settings where the associated unperturbed chain is geometrically ergodic (but not necessarily uniformly ergodic). Our bounds on the difference between the $n$-th step distributions of the perturbed and the unperturbed chains require two inputs: First, we need to control the error made in approximating the likelihood, at least in the center of the state-space. Second, we need to verify a stability condition on the perturbed chain, either by proving a Lyapunov-type condition, or by restricting the perturbed chain to the center of the state space. Joint with Felipe Medina-Aguayo and Daniel Rudolf.