Engineering design is the process through which people draft, evaluate and optimize products, and it forms the foundation of manufacturing. Modern engineering design increasingly demands automated, scalable methods that handle complex geometries and high-dimensional design spaces while accelerating the entire process. This thesis introduces a series of deep geometric learning techniques that provide flexible, differentiable shape representations and enable controllable design exploration, focusing on aerodynamic shape design and optimization. First, I develop a Latent Space Model (LSM) for airfoil Computational Fluid Dynamics (CFD) meshes, an auto-decoder network that encodes each mesh into a low-dimensional latent code and reconstructs it by deforming a fixed template. A novel regularization ensures smooth, valid deformed CFD meshes. Because the mesh geometry remains fully differentiable with respect to the latent code, the LSM allows efficient gradient-based optimization directly through the mesh. Next, I introduce and compare two parameterization strategies. The LSM learns deformation from a database of shapes, while the Direct Mapping Model (DMM) constructs a parameterization on the fly for a single target geometry without training data. Both models incorporate mesh-regularized deformation to preserve computational mesh quality. The analysis clarifies their trade-offs: LSM leverages data priors whereas DMM is data-independent, and each can be chosen based on specific needs. Building on these insights, I develop DeepGeo that generalizes DMM to complex 3D geometries by automatically learning deformation in high-dimensional spaces, thereby streamlining the shape parameterization in design optimization. It provides large deformation freedom while inherently enforcing global smoothness. In case studies, including a 2D circle, the NASA Common Research Model wing, and a Blended-Wing-Body aircraft, DeepGeo produces optimized shapes with aerodynamic performance on par with state-of-the-art hand-crafted parameterizations, while significantly reducing manual effort. The thesis then shifts to design space exploration, a necessary step before design optimization for concept prototyping. I propose DiffGeo, a latent-space diffusion model that generates diverse, valid aerodynamic shapes. By training the diffusion model on the latent codes learned with LSM, DiffGeo achieves high data efficiency and guarantees validity of generated designs. DiffGeo also demonstrates conditional sampling: by guiding the diffusion process with complex geometric objectives, engineers can generate shapes meeting specified criteria. Finally, I participated in developing Dflow-SUR, a physics-guided diffusion approach with high computational efficiency and superior controllability. Optimizing the sampled noises through gradient feedback from a surrogate model produces airfoils and wings with improved aerodynamic performance. These contributions together form a coherent design pipeline: from efficient design space exploration to highly automated design optimization empowered by fully differentiable parameterization models. The results show that deep geometric learning can handle challenging design representations and greatly accelerate the design cycle by automating much of the geometry handling and providing rapid concept generation. This interdisciplinary research paves the way toward more exploratory, efficient and human-AI collaborative design workflows.