Solvability of Cauchy's problem in R2 for subcritical quasi-geostrophic equation is discussed here in two phase spaces; Lp(R2) with p>22α−1 and Hs(R2) with s>1. A solution to that equation in critical case is obtained next as a limit of the Hs-solutions to subcritical equations when the exponent α of (−Δ)α tends to 12+. Such idea seems to be new in the literature. Existence of the global attractor in subcritical case is also studied in the paper. In Section 7 we discuss solvability of the critical problem with Dirichlet boundary condition in a bounded domain Ω⊂R2, when ‖θ0‖L∞(Ω) and ‖f‖L∞(Ω) are small.