We discuss the interplay between superconductivity and formation of bound pairs of fermions (BCS-BEC crossover) in a 2D model of interacting fermions with small Fermi energy ϵF and a weak attractive interaction, which extends to energies well above ϵ_F. The 2D case is special because a two-particle bound state forms at arbitrary weak interaction, and already at weak coupling, one has to distinguish between the bound-state formation and superconductivity. We briefly review the situation in the one-band model and then consider two different two-band models: one with one hole band and one electron band and another with two hole or two electron bands. At ϵ_F≫E_0, where E_0 is the energy of the bound state of two particle in vacuum, the two models behave similar. But in the opposite case they behave differently: in the model with two hole/two electron bands, the instability temperature of formation of cooper pairs T_ins∼E_0/(logE_0/ϵ_F ), and the true superconducting critical temperature T_c∼ϵ_F, like in the one-band model. In between T_ins and T_c, the system displays a preformed pair behavior. In the model with one hole and one electron bands, T_c remains of order T_ins, and both remain finite at ϵ_F=0 and of the order of E_0. The preformed pair behavior still does exist in this model because T_c is numerically smaller than T_ins.