  The phase transition to a Bose-Einstein condensate is unusual in that it is
not necessarily driven by inter-particle interactions but can occur in an ideal
gas as a result of a purely statistical saturation of excited states. However,
interactions are necessary for any system to reach thermal equilibrium and so
are required for condensation to occur in finite time. In this Chapter we
review the role of interactions in Bose-Einstein condensation, covering both
theory and experiment. We focus on measurements performed on harmonically
trapped ultracold atomic gases, but also discuss how these results relate to
the uniform-system case, which is more theoretically studied and also more
relevant for other experimental systems.
  We first consider interaction strengths for which the system can be
considered sufficiently close to equilibrium to measure thermodynamic
behaviour. In particular we discuss the effects of interactions both on the
mechanism of condensation (namely the saturation of the excited states) and on
the critical temperature at which condensation occurs. We then discuss in more
detail the conditions for the equilibrium thermodynamic measurements to be
possible, and the non-equilibrium phenomena that occur when these conditions
are controllably violated by tuning the strength of interactions in the gas.
