The Art Gallery Problem consists of determining the minimum number of guards required to guard a gallery whose floor plan corresponds to a polygon with \(n\) vertices. Vaclav Chvatal was the first to prove that \(\lfloor \frac{n}{3}\rfloor\) guards will always be sufficient to guard a polygon with \(n\) vertices. The primary goal of this thesis is to thoroughly examine this result, i.e. Chvatal’s theorem. The first part of the thesis presents the fundamental concepts, results and several observations regarding polygon triangulation, which is a crucial part of Chvatal’s theorem. The text proceeds with Chvatal’s proof, followed by Fisk’s proof of the same result based on the coloring of the polygon vertices. In the second part of the thesis, several modifications of the original problem and a detailed analysis of a special type of polygon, the orthogonal polygon, are presented. A proof for the claim that guarding an orthogonal polygon with \(n\) vertices will never require more than \(\lfloor \frac{n}{4}\rfloor\) stationary guards is provided in the subsequent section. At the very end of the thesis, galleries in three-dimensional space are considered, and the essential observations related to guarding an arbitrary polyhedron are presented.