The dynamics of soliton and quasisoliton solutions of the cubic third-order nonlinear Schrodinger equation is studied. Regular solitons exist due to a balance between the nonlinear terms and (linear) third-order dispersion; they are not important at small alpha (3) (alpha (3) is the coefficient in the third derivative term) and vanish at alpha3 -->0. The most essential, at small alpha (3), is a quasisoliton emitting resonant radiation (resonantly radiating soliton). Its relationship with the other (steady) quasisoliton, called embedded soliton, is studied analytically and also in numerical experiments. It is demonstrated that the resonantly radiating solitons emerge in the course of nonlinear evolution, which shows their physical significance.