This Master’s Thesis with the subject of polyhedra contains an introductory chapter with relevant definitions and the main part of the work consisting of four chapters. In the second chapter, we show constructively that regular or Platonic solids do exist. In order to make them unique, we amend Euclid’s definition of regular polyhedra adding the condition of convexity and the requirement that an equal number of sides meet in each vertex. The two most important results are given in the third chapter. Euler’s theorem describes the relationship between the number of vertices, edges and sides of a convex polyhedron, while Cauchy’s theorem explains when two convex polyhedra are congruent. Applying these theorems, we determine all convex deltahedra, i.e. polyhedra whose all sides are equilateral triangles. Convex polyhedra with sides that are regular polygons are classified in the fourth chapter where we distinguish between uniform and non-uniform polyhedra. Semiregular polyhedra, being uniform polyhedra, contain regular polyhedra, prisms, antiprisms and Archimedean solids. We also show that there are only finitely many non-uniform convex polyhedra with regular sides. In the final chapter, we investigate symmetry groups of regular polyhedra. Their rotation groups, together with cyclic and dihedral groups, are the only finite subgroups of the rotation group of a sphere.