We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. ÍøÂç²ÊƱAPPÏÂÔØ_°Ä¿Í²ÊƱÍø-¹Ù·½ÓÎÏ·, approach introduces local approximations of these models into the Metropolis-Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler's exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the \emph{exact} posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. ÍøÂç²ÊƱAPPÏÂÔØ_°Ä¿Í²ÊƱÍø-¹Ù·½ÓÎÏ·, theoretical results reinforce the key observation underlying this paper: when the likelihood has some \emph{local} regularity, the number of model evaluations per MCMC step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ODE and PDE inference problems, with both synthetic and real data.