The problem of maneuvering target tracking is addressed in this paper. The main challenge in maneuvering target tracking stems from the nonlinearity and non-Gaussianity of the problem. The Singer model was used to model the maneuvering target dynamics and abrupt changes in the acceleration. According to this model, the heavy-tailed Cauchy distribution driving noise is used to model the abrupt changes in the target acceleration. The nonlinear, non-Gaussian Kalman filter was applied to this problem. The algorithm is based on the Gaussian mixture model for the posterior state vector. The nonlinear, non-Gaussian Kalman filter for this problem was tested using simulations, and it is shown that it outperforms both the particle filter and the extended Kalman filter.