Wavelet analysis is a rapidly developing area of mathematical and application-oriented research in many disciplines of science and engineering. The wavelet transform is a localized transform in both space (time) and frequency, and this property can be advantageously used to extract information from a signal that is not possible to unravel with a Fourier or even windowed Fourier transform. Wavelet transforms originated in geophysics in early 1980's for the analysis of seismic signal. After a decade of significant mathematical formalism they are now also being exploited for the analysis of several other geophysical processes such as atmospheric turbulence, space-time rainfall, ocean wind waves, seafloor bathymetry, geologic layered structures, climate change, among others. Due to their unique properties, well suited for the analysis of natural phenomena, it is anticipated that there will be an explosion of wavelet applications in geophysics in the next several years. This chapter provides a basic introduction to wavelet transforms and their most important properties. The theory and applications of wavelets is developing very rapidly and we see this chapter only as a limited basic introduction to wavelets which we hope to be of help to the unfamiliar reader and provide motivation and references for further study.