This paper introduces a family of wavelet transforms based on the Poisson Transform. The wavelet transform maps L-2(R) functions to a space described by two continuous variables, scale and translation, as well as a discrete index. Reconstruction in the wavelet domain can be done for each of the discrete indices. Additionally, a different reconstruction formula exists for the Poisson Transform domain. We develop the Poisson Wavelet Transform, present an example relevant to stable, over-damped, linear, time-invariant systems, and show the relationship between the Poisson Transform and the Poisson Wavelet Transform.