Wigner’s idea, that to each elementary particle is associated an irreducible representation of the Poincare group gives fundamental importance to these representations. They are non-trivial mathematical objects. We strive to give a mathematically sound and complete description of the physically relevant representations, and the multiple ways they can be presented, while avoiding the pitfall of relying on advanced representation theory. The representations corresponding to massive particles depend, besides the mass, on a single non-negative integer which corresponds to the spin of the particle. The representations which correspond to massless particles depend on an integer, the helicity, which is a property somewhat similar to the spin. We investigate that action of parity and the operation of taking a “mirror image” of a particle. Finally we provide a brief account of Dirac’s equation.