First, are discussed “Widom’s insertion method” and the “deletion” and “insertion/deletion” methods of Shing and Gubbins. In spite of the potential of the deletion mechanism to handle dense systems, its results were found generally inferior to “insertion.” Therefore, several attempts were made to improve the deletion process, among them is Personage’s technique, which can also be derived from Bennett’s formula. Kofke and Cummings (KC) suggested three more formulas related to Personage’s method, among them, “a staged deletion” and “a staged insertion”; KC performed preliminary MC studies of a (modified) staged insertion where a hard sphere was introduced to the LJ system. Following this idea, Boulougouris et al. carried out a more extensive study of the Bennett formula for the LJ potential, adding a hard sphere within the NPT ensemble. Their method “the staged particle deletion” was extended also to polymers and was found to be superior to Widom’s technique. Another approach is “the ideal gas gauge method” (IGGC) due to Neimark and Vishnyakov. Thus, an LJ fluid system of volume, V at T is in a chemical equilibrium with a gauge system of ideal gas at the same T but volume, Vg . IGGC is a manifestation of the “mesoscopic canonical ensemble” – an intermediate between the canonical and the grand canonical ensembles. Methods for polymers are “scanning” (Meirovitch) and the “incremental chemical potential” of Kumar et al., where a chain is inserted to a system monomer by monomer. To this category pertains “the expanded variable-length chain method” of Escobedo and de Pablo. Finally, thermodynamic integration is a robust method for polymers as demonstrated by Müller and Paul and Wilding and Müller.