The establishment of new spectral classes cooler than type M has had a brief, yet already rich, history. Prototypes of the new "L dwarf" and "T dwarf" classes were first found in the late 1980s to mid-1990s, with a flood of new discoveries occurring in the late 1990s with the advent of deep, large-area, digital sky surveys. Over four hundred and fifty L and T dwarfs are now catalogued. This review concentrates on the spectroscopic properties of these objects, beginning with the establishment of classification schemes rooted in the MK Process. The resulting grid of spectral types is then used as a tool to ferret out the underlying physics. The temperature ranges covered by these spectral types, the complex chemical processes responsible for the shape of their emergent spectra, their nature as either true stars or brown dwarfs, and their number density in the Galaxy are discussed. Two promising avenues for future research are also explored: the extension of the classification system to three dimensions to account for gravity- and metallicity-dependent features, and the capability of newer large-area surveys to uncover brown dwarfs cooler than those now recognized.