Conventional and Non-Conventional Condensations of Bosons

We present a review of some published results concerning evolution of the notion of (conventional ) Bose-Einstein condensation (BEC) predicted in Ideal Bose-Gas (IBG) in 1925. The first step forward was the notion of the conventional generalised BEC à la van den Berg-Lewis-Pulé discovered in the IBG and later in certain models with interactions in 1980s. The next development was a discovery of non-conventional condensation of bosons that occurs only due to interaction. Finally we discuss an example of (non-ideal) the Weakly Imperfect Bose-Gas (WIBG), where the conventional and non-conventional condensates may coexist due to interaction.