Reconstructing models of the real world, including 3D geometry, appearance, and motion of real scenes, is essential for computer graphics and computer vision. It enables the synthesizing of photorealistic novel views, useful for the movie industry and AR/VR applications. It also facilitates the content creation necessary in computer games and AR/VR by avoiding laborious manual design processes. Further, such models are fundamental for intelligent computing systems that need to interpret real-world scenes and actions to act and interact safely with the human world. Notably, the world surrounding us is dynamic, and reconstructing models of dynamic, non-rigidly moving scenes is a severely underconstrained and challenging problem. This state-of-the-art report (STAR) offers the reader a comprehensive summary of state-of-the-art techniques with monocular and multi-view inputs such as data from RGB and RGB-D sensors, among others, conveying an understanding of different approaches, their potential applications, and promising further research directions. The report covers 3D reconstruction of general non-rigid scenes and further addresses the techniques for scene decomposition, editing and controlling, and generalizable and generative modeling. More specifically, we first review the common and fundamental concepts necessary to understand and navigate the field and then discuss the state-of-the-art techniques by reviewing recent approaches that use traditional and machine-learning-based neural representations, including a discussion on the newly enabled applications. The STAR is concluded with a discussion of the remaining limitations and open challenges.