An important challenge in visual Simultaneous Localisation and Mapping (SLAM) has been on the design of scene representations that allow for both robust inference and useful interaction. The rapid progression of semantic image understanding powered by deep learning has led to SLAM systems that enrich geometric maps with semantics, which increases the range of applications possible. However, a core challenge remains in how to tightly integrate geometry and semantics for 3D re- construction; we believe that their joint representation is the right direction for actionable and robust maps. In this thesis we will address the central question on designing efficient scene representations by the use of compressive models, which can represent detail with the least number of parameters. We then demonstrate that compressive models offer a solution for the joint representation of geometry and semantics, where semantics provide priors for robust reconstruction and geometric compression informs scene decomposition. This work focuses on using generative neural networks, a category of compressive representations, for incremental dense SLAM. We develop a volumetric rendering formulation for the use of compressive models in generative inference from multi- view images, enabling two novel SLAM systems. First, we learn class-level code descriptors for object shape from aligned 3D models. At test time, the code and object pose are optimised for efficient and complete object reconstruction from in- stances of the learned categories. This method relaxes the assumption of fixed tem- plates and allows for intra-class shape variation. We demonstrate the usefulness of semantic priors for complete and precise reconstruction in a robotic packing applic- ation. Second, we present a scene-specific multi-layered perceptron (MLP) neural field for full generative dense SLAM. Our results show that it allows for efficient mapping, automatic hole-filling, and joint optimisation of camera trajectory and 3D map. Last, we demonstrate that the MLP’s automatic scene compression discovers underlying scene structures that are revealed with sparse labeling.