The concept of schemes was introduced in 1960 by Alexander Grothendieck as an attempt to unify many different approaches to some of the most important questions of algebraic geometry. In the meantime, the scheme-theoretic approach proved very efficient, and schemes became a standard feature of algebraic geometry as well as other related disciplines. The goal of this paper was to provide an exposition of the fundamentals of scheme theory and to establish a connection between the newly introduced concepts and their counterparts in the classical theory of projective varieties. In the opening chapter we introduced the notions we would ultimately generalize using schemes: affine, projective, and quasi-projective varieties. We also commented on some of the more important properties of varieties and stated some basic results concerning them. In the following chapters we described the key components of schemes: the spectrum of a ring and its associated structure sheaf. In the second chapter we observed quite a few examples of spectra; by introducing the Zariski topology, we made the spectrum into a topological space and discussed its topological similarities and differences with respect to the classical affine varieties. The main content of the third chapter was the introduction of (pre-)sheafs as well as the proof of the fact that the structure presheaf is indeed a sheaf. These concepts enabled us to define a scheme as a ringed space which ”locally looks like” the spectrum of a ring. We also studied morphisms of schemes and some basic constructions, such as gluing and closed subschemes. An important part of the fourth chapter consisted of describing the way in which we can regard any quasi-projective variety as a scheme. Finally, in the last chapter, we studied in detail some of the most important properties of schemes. We concluded the paper by defining algebraic varieties and linking the described properties to some familiar results of the classical theory of projective varieties.