For some time it has been known that the continuous Heisenberg-Weyl group provides a theoretical basis for radar detection. However, it has recently become apparent that the finite multidimensional Heisenberg-Weyl groups also have a role to play. In this talk we will describe the theory of the Heisenberg-Weyl groups in relation to the development of waveform libraries for adaptive radar. These groups form the basis for the representation of the radar environment in terms of operators on the space of waveforms, as well as a unified basis for the construction of useful waveform/sequence libraries for radar. We will emphasize the Golay complementary or Welti waveforms/sequences and show that their existence and properties can be understood in terms of the relationship between two different finite Heisenberg-Weyl groups. We will also discuss how within the Golay sequences of length 2^m there exist pairs of sequences which are complementary in the squares of their autocorrelation function and how these pairs have a novel application to radar detection.