A family of multivariate dynamic generalized linear models is introduced as a general framework for the analysis of time series with observations from the exponential family. Besides common conditionally Gaussian models, this article deals with univariate models for counted and binary data and, as the most interesting multivariate case, models for nonstationary multicategorical time series. For univariate responses, a related yet different class of models has been introduced in a Bayesian setting by West, Harrison and Migon. Assuming conjugate prior-posterior distributions for the natural parameter of the exponential family, they derive an approximate filter for estimation of time-varying states or parameters. However, their method raises some problems; in particular, in extending it to the multivariate case. A different approach to filtering and smoothing is chosen in this article. To avoid a full Bayesian analysis based on numerical integration, which becomes computationally critical for higher dimensions, we propose to estimate time-varying parameters by posterior modes. A generalization of the extended Kalman filter and smoother for conditionally Gaussian observations is suggested for approximate posterior mode estimation. For the purpose of comparison, it is applied to data sets analyzed by the authors mentioned earlier. The quality of approximation is also studied by simulation experiments, indicating good estimation behavior, and an application to multicategorical business test data is given.