In paper , Petru T. Mocanu has obtained suﬃcient conditions for a function in the classes C1 (U), respectively C2 (U) to be univalent and to map U onto a domain which is starlike (with respect to origin), respectively convex. Those conditions are similar to those in the analytic case. In paper , Petru T. Mocanu has obtained suﬃcient conditions of univalency for complex functions in the class C1 which are also similar to those in the analytic case. Having those papers as inspiration, we have tried to introduce the notion of subordination for non-analytic functions of classes C1 and C2 following the classical theory of diﬀerential subordination for analytic functions introduced by S.S. Miller and P.T. Mocanu in papers  and  and developed in the book . Let Ω be any set in the complex plane C, let p be a non-analytic function in the unit disc U, p ∈ C2(U) and let ψ(r, s, t; z) : C3×U → C. In article  we have considered the problem of determining properties of the function p, non-analytic in the unit disc U, such that p satisﬁes the diﬀerential subordination. ψ(p(z), Dp(z), D2p(z) − Dp(z); z) ⊂ Ω ⇒ p(U) ⊂ ∆. The present chapter is based on the results contained in paper , some parts of it have been removed and results obtained after the appearance of the paper have been added.