The first attempts at solving a binary black hole spacetime date back to the 1960s, with the pioneering works of Hahn and Lindquist. In spite of all the computational advances and enormous efforts by several groups, the first stable, long-term evolution of the orbit and merger of two black holes was only accomplished over 40 years later, in 2005. Since then, the field of Numerical Relativity has matured, and been extensively used to explore and uncover a plethora of physical phenomena in various scenarios. In this thesis, we take this field to new frontiers by exploring its extensions to higher dimensions, non-asymptotically flat spacetimes and Einstein-Maxwell theory. We start by reviewing the usual formalism and tools, including the "3+1" decomposition, initial data construction, the BSSN evolution scheme and standard wave extraction procedures. We then present a dimensional reduction procedure that allows one to use existing numerical codes (with minor adaptations) to evolve higher-dimensional systems with enough symmetry, and show corresponding results obtained for five-dimensional head-on collisions of black holes. Finally, we show evolutions of black holes in non-asymptotically flat spacetimes, and in Einstein-Maxwell theory.