  One of the most exciting potential sources of gravitational waves for the
Laser Interferometer Space Antenna (LISA) are the inspirals of approximately
solar mass compact objects into massive black holes in the centres of galaxies
- extreme mass ratio inspirals (EMRIs). LISA should observe between a few tens
and a few hundred EMRIs over the mission lifetime, mostly at low redshifts (z <
1). Each observation will provide a measurement of the parameters of the host
system to unprecendented precision. LISA EMRI observations will thus offer a
new and unique way to probe black holes at low redshift. In this article we
provide a description of the population of EMRI events that LISA is likely to
observe, and describe how the numbers of events vary with changes in the
underlying assumptions about the black hole population. We also provide fitting
functions that characterise LISA's ability to detect EMRIs and which will allow
LISA event rates to be computed for arbitrary population models. We finish with
a discussion of an ongoing programme that will use these results to assess what
constraints LISA observations could place on galaxy evolution models.
