This course covers a range of topics in first and last passage percolation. First passage percolation is a model of the shortest path through a random environment and is conjectured to be in the KPZ universality class. However, many of the simplest questions remain open. The tools to study it come mainly from percolation theory. By contrast, several variants of last passage percolation (roughly the longest oriented path in a random environment) are exactly solvable meaning many quantities can be calculated explicitly. We examine recent developments in both settings.