With the rapid development of electronic technology, the power consumption of electronic devices has decreased significantly. Consequently, there is substantial interest in harvesting energy from ambient sources, such as vibration, in order to power small-scale wireless devices. To design optimal vibration harvesting systems it is important to determine the maximum power obtainable from a given vibration source. Initially, white noise base excitation of a general nonlinear energy harvester model is considered. The power input from white noise is known to be proportional both to the total oscillating mass of the system and the magnitude of the noise spectral density, regardless of the internal mechanics of the system. This power is split between undesirable mechanical damping and useful electrical dissipation, where the form of the stiffness profile and device parameters determine the relative proportion of energy dissipated by each mechanism. An upper bound on the electrical power is derived and used to guide towards optimal harvesting devices, revealing that low stiffness systems exhibit maximum performance. Many engineering applications will exhibit more complicated spectra than the flat spectrum of white noise. Expanding upon the white noise analysis, a method to investigate the power dissipation of nonlinear oscillators under non-white excitation is developed by extending the Wiener series. The relatively simple first term of the series, together with the excitation spectrum, is found to completely define the power dissipated. An important property of this first term, namely that the integral over its frequency domain representation is proportional to the oscillating mass, is derived and validated both numerically and experimentally, using a base excited cantilever beam with a nonlinear restoring force produced by magnets. Another form of excitation prevalent in many mechanical systems is a combination of deterministic and broadband random vibration. Lastly, the Duffing oscillator is used to illustrate the behaviour of a nonlinear system under this form of excitation, where the response is observed to spread around the attractor that would be seen if purely deterministic excitation was present. The ability of global weighted residual methods to produce the complex responses typical of nonlinear oscillators is assessed and found to be accurate for systems with weak nonlinearity.