The physically plausible simulation of objects is an active area of research in computer graphics. It is used in movies and video games and to simulate phenomena that would be expensive to do in a laboratory. In the traditional simulation method, external and internal forces acting on an object are used to calculate acceleration using Newton’s second law; then a time integration method is used to first update the velocity and then the position of the object. A recent alternative is to use the position-based dynamics (PBD) algorithm that removes the acceleration and velocity layers and directly calculates the position satisfying constraint functions. The main advantages of PBD over the traditional method are that it is faster, it is controllable, and it avoids the problem of overshooting due to explicit integration in time. Its main disadvantages are the lack of effective convergence and its dependence on a stiffness factor *k*, the timestep *Δt*, and the number of iterations *n*, which causes insufficient accuracy.

The objective of this paper is to review the improvements and applications of the PBD algorithm since 2018, including extensions for different materials and integration with other techniques. The paper is divided into four sections: a description of the algorithm, recent improvements, recent applications, and future research.

The first section presents a complete and comprehensive description of the basic algorithm. The second section describes the latest improvements. This part describes the hierarchical PBD (HPDB) algorithms and the application of second-order backward differentiation (BDF2) to improve convergence efficiency, as well as extended PBD (XPBD) and projective dynamics (PD) to eliminate dependencies in the factors *k*, *Δt*, and *n*. It also mentions that PBD was initially used to simulate particle-based objects, but that its use has now been extended to simulations of rigid bodies, fluids, and clothes. The third section describes recent applications of PBD related to deep learning, medicine, and architecture. The fourth section suggests some future research directions: fully resolving the limitations of the algorithm, integration with other computational methods, integration with software packages, integration with extended reality, use of cloud computing, and improvements in volumetric models.

In a final evaluation, this paper can be useful to both those interested in learning about the PBD algorithm and those interested in updating and learning about the latest improvements and applications of the algorithm. The paper is very well written and perhaps its greatest contribution is the tables that summarize the main contributions and characteristics of all the papers that the authors reviewed.