In this talk, we will be discussing the discrete fourier transform in one and two dimensions, including some of the transform's properties, as well as various strategies for efficiently evaluating the discrete fourier transform via several fast fourier transform algorithms. The algorithms discussed will include the Cooley-Turkey algorithm, the radix-2 decimation in time strategy, the split-radix algorithm, the mixed-radix algorithm, the prime factor algorithm, and Rader's algorithm. Finally, we will discuss various applications of the fast fourier transform, as well as considerations for using it in practical applications.