Not sure where Monte Carlo came from on this thread, and I had not previously thought of it as "Maybe the most brilliant mathematical modeling protocol ever invented in relation to physical science or ever invented, period" but I do recall falling in love with it when I first discovered it. Like Constantinos, I knew it as a technique for calculating definite integrals, but it is almost infinitely flexible. I once needed to model large polymer molecules adhering to metallic surfaces where the sticking coefficient was unknown. With Monte Carlo methods one can very easily 'shape' the distribution, and it was known that the particles re-evolved with a cosine distribution. This was easy to solve analytically in 2D so I both Monte Carlo modeled the process and also calculated the exact solution and they agreed as closely as I wished (just run longer!) Thus convinced of the validity of the technique I used the Monte Carlo in a 3D configuration (which I could not solve) and got great results to compare with experiment. I suspect there are thousands of unique applications of Monte Carlo to accomplish otherwise almost impossible tasks.
Edwin Eugene Klingman