The number of limit cycles and the bifurcation are considered for the nonlinear system (1) \(\dot x=P(x,y)\), \(\dot y=Q(x,y)\) where P and Q are cubic polynomials in x and y. A number of classes of (1) with several limit cycles are given. Sufficient conditions for the existence of six small-amplitude limit cycles are established. A class of (1) in which a group of small amplitude limit cycles is surrounded by a large amplitude limit cycle of the opposite orientation is described. Simultaneous bifurcation from several critical points is also considered.