Over the past decade, the advent of deep learning and large-scale computing has immeasurably changed the ways we process, interpret, and predict with data in imaging and computer vision. The ``traditional'' approach to algorithm design, based around parametric models for specific structures of signals and measurements---say sparse and low-rank models---and the associated optimization toolkit, is now significantly enriched with data-driven learning-based techniques, where large-scale networks are pre-trained and then adapted to a variety of specific tasks. Nevertheless, the successes of both modern data-driven and classic model-based paradigms rely crucially on correctly identifying the low-dimensional structures present in real-world data, to the extent that we see the roles of learning and compression of data processing algorithms---whether explicit or implicit, as with deep networks---as inextricably linked. As such, this tutorial provides a timely tutorial that uniquely bridges low-dimensional models with deep learning in imaging and vision. This tutorial will show how (i) these low-dimensional models and principles provide a valuable lens for formulating problems and understanding the behavior of modern deep models in imaging and computer vision, and (ii) how ideas from low-dimensional models can provide valuable guidance for designing new parameter efficient, robust, and interpretable deep learning models for computer vision problems in practice. The tutorial will start by introducing fundamental low-dimensional models (e.g., basic sparse and low-rank models) with motivating computer vision applications. Based on these developments, we will discuss strong conceptual, algorithmic, and theoretical connections between low-dimensional structures and deep models, providing new perspectives to understand state-of-the-art deep models in terms of learned representations and generative models. Finally, we will demonstrate that these connections can lead to new principles for designing deep networks and learning low-dimensional structures in computer vision, with both clear interpretability and practical benefits.