This thesis focuses on semidirect and Zappa-Szép products in the context of semigroups and monoids. We present a survey of direct, semidirect and Zappa-Szép products and discuss correspondence between external and internal versions of these products for semigroups and monoids. Particular attention in this thesis is paid to a wide class of semigroups known as restriction semigroups. We consider Zappa-Szép product of a left restriction semigroup S with semilattice of projections E and determine algebraic properties of it. We prove that analogues of Green's lemmas and Green's theorem hold for certain semigroups where Green's relations R, L, H and D are replaced by R̃_E, L̃_E, H̃_E and D̃_E. We show that if H̃_E is a congruence on a certain semigroup S, then any right congruence on the submonoid H̃^e_E (the H̃_E-class of e), where eΕE, can be extended to a congruence on S. We introduce the idea of an inverse skeleton U of a semigroup S and examine some conditions under which we obtain skeletons from monoids. We focus on a result of Kunze [37] for the Bruck-Reilly extension BR(M,θ) of a monoid M, showing that BR(M,θ) is a Zappa-Szép product of N⁰ under addition and a semidirect product M x \N⁰. We put Kunze's result in more general framework and give an analogous result for certain restriction monoids. We consider the λ-semidirect product of two left restriction semigroups and prove that it is left restriction. In the two sided case using the notion of double action we prove that the λ-semidirect product of two restriction semigroups is restriction. We introduce the notion of (A,T)-properness to prove the results analogous to McAlister's covering theorem and O'Carroll's embedding theorem for monoids and left restriction monoids under some conditions. We extend the notion of the λ-semidirect product of two restriction semigroups S and T to develop λ-Zappa-Szép products and construct a category. In the special case where S is a semilattice and T is a monoid we order our category to become inductive and thus obtain a restriction semigroup via the use of the standard pseudo-product.