This paper explores the use of deep-learning frameworks both for imitation learning (supervised) and unsupervised methods applied to structure classification and understanding in CR3BP dynamics typically arising in astrodynamics. More specifically, the goal is to explore the use of deep architectures such as Convolutional Neural Networks (CNN) and Variational Autoencoders (VAE) to classify families of periodic orbits within and across CR3BP dynamics (e.g. Earth-Sun, Jupiter-Sun). It is demonstrated that CNNs are capable of capturing non-linear decision boundaries that enable distinctions between family of orbits. Importantly, VAEs are designed to model the distribution of data comprising periodic orbits. The key distributions parameters (mean and variance) are modeled using neural networks. Such parameters are then plotted on 2-D graph to visually evaluate the clustering of the orbits which enables classification. VAEs are also compared with non-linear, statistically-based dimensionality reduction methods (e.g. t-Distributed Stochastic Neighbor Embedding or t-SNE) that project trajectory data in lower-dimensional embedding, while preserving distance metrics. It is shown that individual families generate clusters that can be easily and visually distinguished in a 2-D plane. This study is an initial attempt to employ the features learned by deep networks in a data-driven fashion to better understand and identify dynamical structure arising in astrodynamics.