There is a family of related methods, collectively known as MCMC methods. These stochastic methods allow us to get samples from the true posterior distribution as long as we are able to compute the likelihood and the prior point-wise. While this is the same condition that we need for the grid-approach, MCMC methods outperform the grid approximation. The is because MCMC methods are capable of taking more samples from higher-probability regions than lower ones. In fact, an MCMC method will visit each region of the parameter-space in accordance to their relative probabilities. If region A is twice as likely as region B, then we are going to get twice as many samples from A as from B. Hence, even if we are not capable of computing the whole posterior analytically, we could use MCMC methods to take samples from it.
At the most fundamental level, basically everything...