Dynamic Linear Models (DLMs) are commonly employed for time series analysisdue to their versatile structure, simple recursive updating, ability to handlemissing data, and probabilistic forecasting. However, the options for counttime series are limited: Gaussian DLMs require continuous data, whilePoisson-based alternatives often lack sufficient modeling flexibility. Weintroduce a novel semiparametric methodology for count time series by warping aGaussian DLM. The warping function has two components: a (nonparametric)transformation operator that provides distributional flexibility and a roundingoperator that ensures the correct support for the discrete data-generatingprocess. We develop conjugate inference for the warped DLM, which enablesanalytic and recursive updates for the state space filtering and smoothingdistributions. We leverage these results to produce customized and efficientalgorithms for inference and forecasting, including Monte Carlo simulation foroffline analysis and an optimal particle filter for online inference. Thisframework unifies and extends a variety of discrete time series models and isvalid for natural counts, rounded values, and multivariate observations.Simulation studies illustrate the excellent forecasting capabilities of thewarped DLM. The proposed approach is applied to a multivariate time series ofdaily overdose counts and demonstrates both modeling and computationalsuccesses.