In paper [1], Petru T. Mocanu has obtained suﬃcient conditions for a function in the classes C^{1} (U), respectively C^{2} (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 [2], Petru T. Mocanu has obtained suﬃcient conditions of univalency for complex functions in the class C^{1} 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 C^{1} and C^{2} following the classical theory of diﬀerential subordination for analytic functions introduced by S.S. Miller and P.T. Mocanu in papers [3] and [4] and developed in the book [5]. Let Ω be any set in the complex plane C, let p be a non-analytic function in the unit disc U, p ∈ C^{2}(U) and let ψ(r, s, t; z) : C^{3}×U → C. In article [6] 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), D^{2}p(z) − Dp(z); z) ⊂ Ω ⇒ p(U) ⊂ ∆. The present chapter is based on the results contained in paper [7], some parts of it have been removed and results obtained after the appearance of the paper have been added.