In this chapter we study quantum symmetry in detail. We start from general properties and results (Wigner theorem, Lie algebras, representations, characters, etc.). Then we specialize the discussion to the rotation symmetry in \(\mathbb {R}^3\) , angular momentum, and spin. We construct the \(SU(2)\) representations and explain the spin selection rule. We construct explicitly the spherical harmonics in \(\mathbb {R}^3\) , and then generalize the construction to spherical harmonics in \(\mathbb {R}^n\) for arbitrary n using the method of the harmonic polynomials. These methods are used to solve Quantum Mechanics on spheres \(S^n\) , in symmetric spaces, and other manifolds (with or without boundaries). We construct the \(SU(2)\) rotation operators (in terms of Jacobi polynomials) and then generalize the result to prove the Peter-Weyl theorem for arbitrary compact Lie groups. We outline the Galilei symmetry of non-relativistic Quantum Mechanics and its peculiar features. Then we consider the systems of identical particle and introduce the Bose and Fermi statistics, solve the Fermi oscillator and explain its relation to the half-spin system. In the final two sections we introduce supersymmetry in the Quantum Mechanical set-up: we discuss its representation theory, define the Witten index, and describe its cohomological interpretation.