A new family of wavelets derived from the Poisson distribution is introduced. The associated wavelet transform is closely related to the Poisson transform. The Poisson wavelet transform is characterized by the usual translation and scaling parameters and an additional discrete ordering index. In this paper, the Poisson transform is reviewed, the properties of the Poisson wavelet transform are developed, and two examples are provided to illustrate the utility of this transform for analyzing time delayed signals that are exponentially damped and;for parameter identification.