Goal of this paper is to examine the basic properties of the mean curvature flow of hypersurfaces in Euclidean space. In the first chapter, a brief review of the theory of plane curves is given. In the second chapter, the curve shortening flow is analyzed. Firstly, we defined the curve shortening flow in the form of a Cauchy problem, then the existence and uniqueness theorem for the solution is obtained and the main properties of the flow are studied. Furthermore, we proved the Gage-Hamilton theorem which gives a complete description of the curve shortening flow of convex curves and we showed how Grayson's theorem gives a full description of the curve shortening flow for simply closed curves. The third chapter is used for the review of the basic results regarding the differential and tensor calculus on hypersurfaces as well as the definition of the mean curvature. Finally, in the fourth chapter we introduced the mean curvature flow of a hypersurface in the form of a Cauchy problem. We proved the existence and uniqueness theorem for the solution of the flow as well as some of the basic properties of the flow, for example the avoidance principle. In the end of the chapter, we stated Huisken's theorem which gives a complete description of the mean curvature flow of compact convex hypersurfaces.