Book excerpt: Deep learning has revolutionized the field of 3D shape reconstruction, unlocking new possibilities and achieving superior performance compared to traditional methods. However, despite being the dominant 3D shape representation in real-world applications, polygon meshes have been severely underutilized as a representation for output shapes in neural 3D reconstruction methods. One key reason is that triangle tessellations are irregular, which poses challenges for generating them using neural networks. Therefore, it is imperative to develop algorithms that leverage the power of deep learning while generating output shapes in polygon mesh formats for seamless integration into real-world applications. In this thesis, we propose several data-driven approaches to reconstruct explicit meshes from diverse types of input data, aiming to address this challenge. Drawing inspiration from classical data structures and algorithms in computer graphics, we develop representations to effectively represent meshes within neural networks. First, we introduce BSP-Net. Inspired by a classical data structure Binary Space Partitioning (BSP), we represent a 3D shape as a union of convex primitives, each obtained by intersecting half-spaces. This 3-layer BSP-tree representation allows a shape to be stored in a 3-layer multilayer perceptron (MLP) as a neural implicit, while an exact polygon mesh can be extracted from the MLP weights by parsing the underlying BSP-tree. BSP-Net is the first shape decoder neural network that can produce compact and watertight polygon meshes natively. Next, we present Neural Marching Cubes (NMC), a data-driven algorithm for reconstructing meshes from discretized implicit fields. NMC is built upon Marching Cubes (MC), but it learns the vertex positions and local mesh topologies from example training meshes, thereby avoiding topological errors and achieving better reconstruction of geometric features. In our subsequent work, Neural Dual Contouring (NDC), we replace the MC meshing algorithm with Dual Contouring (DC) with slight modifications, so that our algorithm that can reconstruct meshes from signed/unsigned distance fields, binary voxels, and point clouds, with high accuracy and fast inference speed. Furthermore, inspired by the volume rendering algorithm in Neural Radiance Fields (NeRF), we introduce differentiable rendering to NDC to arrive at MobileNeRF, a NeRF based on triangle meshes with view-dependent textures that can reconstruct scenes from multi-view images. MobileNeRF is the first NeRF-based method that can run on phones and other mobile devices, exhibiting its efficiency and compatibility.