Deep learning techniques have proven effective in many applications, but these implementations mostly apply to data in one or two dimensions. Handling 3D data is more challenging due to its irregularity and complexity, and there is a growing interest in adapting deep learning techniques to the 3D domain. A recent successful approach called MeshCNN consists of a set of convolutional and pooling operators applied to the edges of triangular meshes. While this approach produced superb results in classification and segmentation of 3D shapes, it can only be applied to edges of a mesh, which can constitute a disadvantage for applications where the focuses are other primitives of the mesh. In this study, we propose face-based and vertex-based operators for mesh convolutional networks. We design two novel architectures based on the MeshCNN network that can operate on faces and vertices of a mesh, respectively. We demonstrate that the proposed face-based architecture outperforms the original MeshCNN implementation in mesh classification and mesh segmentation, setting the new state of the art on benchmark datasets. In addition, we extend the vertex-based operator to fit in the Point2Mesh model for mesh reconstruction from clean, noisy, and incomplete point clouds. While no statistically significant performance improvements are observed, the model training and inference time are reduced by the proposed approach by 91% and 20%, respectively, as compared with the original Point2Mesh model.