Abstract: The spin group, $Spin(n)$, for $n>2$, can be defined as the universal covering group of $SO(n)$. They can be explicitly constructed as a subgroup of the group of invertible elements of the Clifford algebra. One can easily see that any irreducible $SO(n)$ representation gives an irreducible representation of $Spin(n)$, however, some irreducible $Spin(n)$ representations cannot be constructed in this way. The main goal of this talk is to construct such representations using the representation theory of Clifford algebras.