  For many years, $^4$He typified Bose-Einstein superfluids, but recent
advances in dilute ultra-cold alkali-metal gases have provided new neutral
superfluids that are particularly tractable because the system is dilute. This
chapter starts with a brief review of the physics of superfluid $^4$He,
followed by the basic ideas of Bose-Einstein condensation (BEC), first for an
ideal Bose gas and then considering the effect of interparticle interactions,
including time-dependent phenomena. Extensions to more exotic condensates
include magnetic dipolar gases, mixtures of two components, and spinor
condensates that require a focused infrared laser for trapping of all the
various hyperfine magnetic states in a particular hyperfine $F$ manifold of
$m_F$ states. With an applied rotation, the trapped BECs nucleate quantized
vortices. Recent theory and experiment have shown that laser coupling fields
can mimic the effect of rotation. The resulting synthetic gauge fields have
produced vortices in a nonrotating condensate.
