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This dissertation presents numerical studies of gravitational waves produced by black holes in two scenarios: perturbations of a single black hole, and the collision of a binary pair. Their detection plays a crucial roll in further testing General Relativity and opens a whole new field of observational astronomy. First, a technique called Cauchy--perturbative matching is revisited in one dimension through the use of new numerical methods, such as high order finite difference operators, constraint-preserving boundary conditions and, most important, a multi-domain decomposition (also referred to as multi-patch, or multi-block approach). These methods are then used to numerically solve the fully non-linear three-dimensional Einstein vacuum equations representing a non-rotating distorted black hole. In combination with a generalization of the Regge-Wheeler-Zerilli formalism, we quantify the effect of the background choice in the wave extraction techniques. It is found that a systematic error is introduced at finite distances. Furthermore, such error is found to be larger than those due to numerical discretization. Subsequently, the first simulations ever of binary black holes with a finite-difference multi-domain approach are presented. The case is one in which the black holes orbit for about twelve cycles before merging. The salient features of this multi-domain approach are: i) the complexity of the problem scales linearly with the size of the computational domain, ii) excellent scaling, in both weak and strong senses, for several thousand processors. As a next step, binary black hole simulations from inspiral to merger and ringdown are performed using a new technique, turduckening, and a standard finite difference, adaptive mesh-refinement code. The computed gravitational waveforms are compared to those obtained through evolution of the same exact initial configuration but with a pseudo-spectral collocation code. Both the gravitational waves extracted at finite locat