In this chapter, we present approximate methods designed for situations where the likelihood function does not have a closed-form expression (e.g., ABC and BSL), or where the sample size and parameter space are large (e.g., INLA and VI), making traditional MCMC and importance sampling (IS) methods ineffective. We provide the theoretical foundations and include applications to illustrate the potential of these methods. Simulation-based algorithms suffer from the curse of dimensionality in the parameter space, while optimization-based approaches typically require explicit evaluation of the likelihood function. To address situations where both challenges arise simultaneously, recent developments, referred to as hybrid methods, combine these strategies with the pseudo-marginal MCMC approach.