  These lecture notes contain an introduction to the physics of quantum
mixtures of ultracold atoms trapped in multiple internal states. I will discuss
the case of fermionic isotopes of alkaline-earth atoms, which feature an
intrinsic SU($N$) interaction symmetry and convenient methods for the optical
manipulation of their nuclear spin. Some research directions will be presented,
with focus on experiments performed in Florence with nuclear-spin mixtures of
$^{173}$Yb atoms in optical lattices.
