  In a spinor Bose-Einstein gas, the non-zero hyperfine spin of the gas becomes
an accessible degree of freedom. At low temperature, such a gas shows both
magnetic and superfluid order, and undergoes both density and spin dynamics.
These lecture notes present a general overview of the properties of spinor
Bose-Einstein gases. The notes are divided in five sections. In the first, we
summarize basic properties of multi-component quantum fluids, focusing on the
specific case of spinor Bose-Einstein gases and the role of rotational symmetry
in defining their properties. Second, we consider the magnetic state of a
spinor Bose-Einstein gas, highlighting effects of thermodynamics and
Bose-Einstein statistics and also of spin-dependent interactions between atoms.
In the third section, we discuss methods for measuring the properties of
magnetically ordered quantum gases and present newly developed schemes for
spin-dependent imaging. We then discuss the dynamics of spin mixing in which
the spin composition of the gas evolves through the spin-dependent interactions
within the gas. We discuss spin mixing first from a microscopic perspective,
and then advance to discussing collective and beyond-mean-field dynamics. The
fifth section reviews recent studies of the magnetic excitations of
quantum-degenerate spinor Bose gases. We conclude with some perspectives on
future directions for research.
