The different methods to calculate cluster membership probabilities

Reliable membership determination is a fundamental step in the study of star clusters. With the advent of $Gaia$ astrometry, a wide range of statistical and machine-learning techniques has been developed to assign membership probabilities. However, the current situation of membership lists is very unsatisfactory. This review summarises the main methodologies, compares their strengths and limitations, and discusses future directions. The aim is to provide a comprehensive overview and to lead to a more efficient and reliable approach for the forthcoming $Gaia$ DR4. Basically, we know of spatial, classical kinematic, and photometric methods, as well as maximum likelihood and Bayesian statistical methods, and machine learning and clustering algorithms. These different methods come with many modifications and flavours. We assessed all the advantages and disadvantages of the known methods to determine cluster membership probabilities. Although nowadays most methods are based on poor statistical numerics, the more robust algorithms should still be taken into account. It is important to apply and compare several methods. The next step must be to define a list of standard star clusters to test and verify all known methods. The list must cover the complete grid of cluster parameters (age, distance, reddening, and metallicity) and total masses.