Bernoulli percolation is a paradigmatic example of a system undergoing a phase transition, with a phase in which there are no large clusters giving way to a phase with a unique large cluster at the critical probability. It is expected that percolation at and near the critical point has a rich, fractal-like mathematical structure, but this remains poorly understood in dimensions 3,4,5, and 6. I give an introduction to and overview of modern percolation theory, with an emphasis on the quantitative understanding of critical percolation on Euclidean lattices.