This paper is concerned with a class of dynamic estimation problems in which the estimator has the ability to dynamically select, from among a temporally evolving set of possibilities, the source of the data on which the estimate will be based. After motivating and formulating this class of attentive estimation problems in some generality, the paper focuses on the special case in which the state of a linear discrete-time dynamical system driven by gaussian noise is to be estimated using linear measurements corrupted by additive gaussian noise. This differs from the standard Kalman filtering problem in that the measurement map at each time step is selectable from a pre-determined set of such maps. When the system dynamics and noise statistics are known, the problem admits a sensor scheduling solution i.e., a criterion for measurement selection that can be used to determine an optimal sequence of output functions in an open-loop fashion prior to the onset of estimation. When the noise statistics or other parameters are unknown, however, closed-loop adaptive strategies for measurement selection can improve estimator performance.