The Elliott SU(3) model is a basic microscopic model of nuclear structure. Its underlying symmetry, based on the three-dimensional harmonic oscillator, plays a central role in the many-fermion nuclear shell-model theory. It provides a fundamental basis for the development of all modern microscopic models of nuclear structure and the construction of standard many-nucleon quantum mechanics of the nucleus. Using the SU(3) Elliott model as an example helps shortly represent the algebraic approach to nuclear structure. Its algebraic structure (Hamiltonian, basis states, dynamical symmetry) is revealed. A special attention is paid to the shell-model representations of the Elliott’s SU(3) dynamical group. Some of its extensions are finally pointed out.