This chapter discusses computational challenges in posterior exploration for inverse problems involving computationally intensive forward models. Inverse problems aim to infer unknown spatial fields x from indirect observations y, where the relationship between x and y is modeled by a physical system. The likelihood function accounts for both model mismatch and sampling errors, and Gaussian errors are assumed for simplicity. The posterior distribution is given by the product of the likelihood and a prior on x, which typically incorporates spatial regularity constraints. The challenge lies in efficiently exploring the posterior using Markov chain Monte Carlo (MCMC) methods, as the high dimensionality of x and the computational cost of the simulator make traditional MCMC approaches impractical. A single-site Metropolis scheme is effective for simple, unimodal posteriors but is inefficient for complex, multimodal posteriors. Multivariate updating schemes, such as the multivariate random walk Metropolis algorithm, can update multiple components of x at once, reducing computational burden. However, these schemes require careful tuning and may not always outperform single-site methods. To address computational challenges, the chapter explores the use of fast, approximate simulators. Two such simulators are introduced: one based on an incomplete multigrid solve and another based on a coarsened representation of the conductivity field. These simulators can be used in delayed acceptance schemes, which reduce the number of calls to the expensive exact simulator. The adaptive multiple step delayed acceptance algorithm is proposed to improve efficiency by incorporating an adaptive error correction term. The chapter also considers Metropolis-coupled MCMC, which uses multiple chains to improve mixing and convergence. The results show that the adaptive multiple step delayed acceptance algorithm significantly improves efficiency, particularly for the EIT application, where the posterior is multimodal and requires careful sampling. The chapter concludes with a discussion of the challenges and potential improvements in MCMC methods for inverse problems, emphasizing the importance of adaptive techniques and the use of fast, approximate simulators.This chapter discusses computational challenges in posterior exploration for inverse problems involving computationally intensive forward models. Inverse problems aim to infer unknown spatial fields x from indirect observations y, where the relationship between x and y is modeled by a physical system. The likelihood function accounts for both model mismatch and sampling errors, and Gaussian errors are assumed for simplicity. The posterior distribution is given by the product of the likelihood and a prior on x, which typically incorporates spatial regularity constraints. The challenge lies in efficiently exploring the posterior using Markov chain Monte Carlo (MCMC) methods, as the high dimensionality of x and the computational cost of the simulator make traditional MCMC approaches impractical. A single-site Metropolis scheme is effective for simple, unimodal posteriors but is inefficient for complex, multimodal posteriors. Multivariate updating schemes, such as the multivariate random walk Metropolis algorithm, can update multiple components of x at once, reducing computational burden. However, these schemes require careful tuning and may not always outperform single-site methods. To address computational challenges, the chapter explores the use of fast, approximate simulators. Two such simulators are introduced: one based on an incomplete multigrid solve and another based on a coarsened representation of the conductivity field. These simulators can be used in delayed acceptance schemes, which reduce the number of calls to the expensive exact simulator. The adaptive multiple step delayed acceptance algorithm is proposed to improve efficiency by incorporating an adaptive error correction term. The chapter also considers Metropolis-coupled MCMC, which uses multiple chains to improve mixing and convergence. The results show that the adaptive multiple step delayed acceptance algorithm significantly improves efficiency, particularly for the EIT application, where the posterior is multimodal and requires careful sampling. The chapter concludes with a discussion of the challenges and potential improvements in MCMC methods for inverse problems, emphasizing the importance of adaptive techniques and the use of fast, approximate simulators.