The use and interest in Monte Carlo (MC) techniques in the field of medical physics have been rapidly increasing in the past years. This is the case especially in particle therapy, where accurate simulations of different physics processes in complex patient geometries are crucial for a successful patient treatment and for many related research and development activities. Thanks to the detailed implementation of physics processes in any type of material, to the capability of tracking particles in 3D, and to the possibility of including the most important radiobiological effects, MC simulations have become an essential calculation tool not only for dose calculations but also for many other purposes, like the design and commissioning of novel clinical facilities, shielding and radiation protection, the commissioning of treatment planning systems, and prediction and interpretation of data for range monitoring strategies. MC simulations are starting to be more frequently used in clinical practice, especially in the form of specialized codes oriented to dose calculations that can be performed in short time. The use of general purpose MC codes is instead more devoted to research. Despite the increased use of MC simulations for patient treatments, the existing literature suggests that there are still a number of challenges to be faced in order to increase the accuracy of MC calculations for patient treatments. The goal of this review is to discuss some of these remaining challenges. Undoubtedly, it is a work for which a multidisciplinary approach is required. Here, we try to identify some of the aspects where the community involved in applied nuclear physics, radiation biophysics, and computing development can contribute to find solutions. We have selected four specific challenges: i) the development of models in MC to describe nuclear physics interactions, ii) modeling of radiobiological processes in MC simulations, iii) developments of MC-based treatment planning tools, and iv) developments of fast MC codes. For each of them, we describe the underlying problems, present selected examples of proposed solutions, and try to give recommendations for future research.