  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.
