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3D reconstruction of deformable (or non-rigid) scenes from a set of monocular 2D image observations is a long-standing and actively researched area of computer vision and graphics. It is an ill-posed inverse problem, since -- without additional prior assumptions -- it permits infinitely many solutions leading to accurate projection to the input 2D images. Non-rigid reconstruction is a foundational building block for downstream applications like robotics, AR/VR, or visual content creation. The key advantage of using monocular cameras is their omnipresence and availability to the end users as well as their ease of use compared to more sophisticated camera set-ups such as stereo or multi-view systems. This survey focuses on state-of-the-art methods for dense non-rigid 3D reconstruction of various deformable objects and composite scenes from monocular videos or sets of monocular views. It reviews the fundamentals of 3D reconstruction and deformation modeling from 2D image observations. We then start from general methods -- that handle arbitrary scenes and make only a few prior assumptions -- and proceed towards techniques making stronger assumptions about the observed objects and types of deformations (e.g. human faces, bodies, hands, and animals). A significant part of this STAR is also devoted to classification and a high-level comparison of the methods, as well as an overview of the datasets for training and evaluation of the discussed techniques. We conclude by discussing open challenges in the field and the social aspects associated with the usage of the reviewed methods.