System \(IL\) is a system of modal logic introduced by Albert Visser in 1988. This system contains one unary modal operator \(\Box\) and one binary modal operator \(\rhd\). We also consider extensions of system \(IL\). They are given by the intepretability principles, and some of these extensions are interpretability logics \(ILM\), \(ILP\), \(ILW\) and \(ILM_0\). Modal completeness of logics \(IL\), \(ILM\) and \(ILP\) was proven in 1990. by De Jongh and Veltman. Later they also proved modal completeness of the logic \(ILW\). In 2004. Evan Goris and Joost Joosten gave new proofs of completeness of logics \(IL\) and \(ILM\) using their step-by-step method. Using that method, they also proved modal completeness of logics \(ILM_0\) and \(ILW^∗\). The main goal of this thesis is to present the proof of modal completeness of interpretability logics \(IL\) and \(ILP\) by using step–by–step method. In the first chapter, we deal with syntax and semantic of the interpretability logic \(IL\). In this chapter we present proof of soundness theorem for logic \(IL\) and proof of deduction theorem for logic \(IL\). After presenting the notion of \(IL\)–consistent and maximal \(IL\)–consistent set, we give proof of Lindenbaum’s lemma. In the second chapter, we define the following notions: labeled frame, adequate frame, quasi–frame, problems, deficiencies, critical cone and generalized cone. We also prove existence lemma and \(IL\)–closure lemma. The main goal of this chapter is to prove model completeness of the logic \(IL\) by using the step–by–step method. In the third chapter, we present the proof of completeness of the logic \(ILP\). We first present definitions and results that are related to interpretability logics \(ILX\). Then we describe main steps of the step–by–step method. Also, we give formal statement of the main lemma. Finally, we present detailed proof of completeness of the logic \(ILP\). We are doing that by checking requirements of the main lemma.