The dynamic linear model or Kalman filtering model provides a useful methodology for predicting the past, present, and future states of a dynamic system, such as an object in motion or an economic or social indicator that is changing systematically with time. Recursive likelihood methods for adaptive Kalman filtering and smoothing are developed. Unknown observational and systematic covariances in the dynamic linear model are estimated recursively using both the EM algorithm and the method of scoring. Such estimates when used in combination with the Kalman filter and fixed interval smoother allow for adaptive filtering and smoothing of state space predictors. In order to assess the performance of the EM and scoring methods for estimation, a small Monte Carlo simulation is performed. Two different types of adaptive filters constructed from the scoring method are described: (1) a fast filter that does not require storage of past data; and (2) a slower filter, that requires past data storage, and is based on the computation of a new maximum likelihood estimate. (PN)