Poisson shot noise is a natural generalization of a compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. This model can be used to model delay in the settlement of insurance claims and we present a statistical data analysis to predict the outstanding liabilities of a car insurance portfolio. We also study the weak limiting behaviour in various regimes with Gaussian and stable limit processes. The shot noise model is also appropriate to model information entering the financial market at Poisson times. Depending on the persistence of this information the limit process is either Brownian motion or Fractional Brownian motion. This leads to an economic interpretation of the observed long range dependence in certain financila data. This talk is based on various papers jointly with Christoph K"uhn, Thomas Mikosch, Annette Sch"arf and Martin Severin.