For Bayesian inference, Markov Chain Monte Carlo (MCMC) methods were a huge breakthrough. These methods provide a principled way for simulating from a posterior probability distribution, and are useful for integrating distributions that are computationally intractable. Usually MCMC methods are performed with computers, but I recently read two papers that apply Monte Carlo simulation in interesting ways.
The first is Markov Chain Monte Carlo with People. MCMC with people is somewhat similar to playing the game of telephone--there is input "data" (think of the starting word in the telephone game) that is transmitted across stages where it can be modified and then output at the end. In the paper the authors construct a task so that human learners approximately follow an MCMC acceptance rule. I have summarized the paper in slightly more detail here.
The second paper is even less conventional: the authors approximate the value of π using a "Mossberg 500 pump-action shotgun as the proposal distribution." Their simulated value is 3.131, within 0.33% of the true value. As the authors state, "this represents the first attempt at estimating π using such method, thus opening up new perspectives towards computing mathematical constants using everyday tools." Who said statistics has to be boring?