We will show the interlacing between complete cotorsion pairs, model structures and homotopy categories. This will give a method of constructing adjoint functors between homotopy categories as well as a method for constructing abelian model structures in the category of unbounded complexes of certain abelian categories. We illustrate our methods by recovering some recents results of Murfet and Neeman as particular instances. And we also find new abelian model structures both in C(R) and in C(Qco(X)) attained to classes which are non necessarily closed under direct limits.