This thesis examines some methods of solution of systems with dynamical symmetry by considering the Coulomb and isotropic potentials on a sphere. It is shown that the classical solutions provide cloned orbits which is a criterion for dynamical symmetry. The extra constants of the motion, which are the generators of the symmetry groups, are found and it is shown that the groups are 80(4) and SUM respectively. However, the form of the commutation relations of the quantum-mechanical operators prevents the direct use of group representation theory. An indirect technique, which Pauli used to solve the usual Coulomb problem, is employed to derive the energy eigenvalues and eigenfunctions of both systems. This technique makes use of the matrix elements of the operators in a basis of energy and angular momentum eigenstates. This is shown to be equivalent to a method of Schrodinger for solving a special class of differential equations. The systems above are generalised to N dimensions and solved by this method. For systems with dynamical symmetry the Schrodinger equation is separable in more than one set of coordinates. This is equivalent to choosing different bases of eigenstates. The sets of coordinates are found and the equations are separated in them but neither they nor the corresponding algebra of matrix elements has been solved.