Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there were finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, many real-world data are not naturally posed in the setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose FunBaT: Functional Bayesian Tucker Decomposition. We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions, and then convert the GPs into state-space priors by constructing equivalent stochastic differential equations (SDE) to reduce computational cost. An efficient inference algorithm is further developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications.