Spatio-temporal data are ubiquitous in the geophysical and environmental sciences, and their study is important for understanding and predicting a wide variety of processes. One of the chief difficulties in modeling spatial processes that change with time is the complexity of the dependence structures that must describe how such a process varies, and the presence of high-dimensional datasets and prediction domains. It is particularly challenging to specify parameterizations for nonlinear dynamical spatio-temporal models that are simultaneously useful scientifically and efficient computationally. Statisticians have developed some "deep" mechanistically-motivated models that can accommodate the uncertainties in the predictions and inference. However,these models can be expensive to run. On the other hand, the machine learning community has been developing more black-box "deep learning" approaches, which can be quite flexible, and in a few cases, can be implemented quite efficiently, but without formal uncertainty quantification. Here, we discuss the pros and cons of these approaches and provide hybrid alternative models at the interface of deep statistical and machine learning.