  In this thesis we consider the data analysis problem of detecting
gravitational waves emitted by inspiraling binary systems. Detection of
gravitational waves will open a new window on the Universe enabling direct
detection of systems such as binary black holes for the first time. In the
first Chapter we show how gravitational waves are derived from Einstein's
General theory of Relativity and discuss the emission of gravitational waves
from inspiraling binaries and how this radiation may be detected using laser
interferometers. Around two thirds of stars inhabit binary systems. As they
orbit each other they will emit both energy and angular momentum in the form of
gravitational waves which will inevitably lead to their inspiral and eventual
merger. To date, searches for gravitational waves emitted during the inspiral
of binary systems have concentrated on systems with non-spinning components. In
Chapter 2 we detail the first dedicated search for binaries consisting of
spinning stellar mass compact objects. We analysed 788 hours of data collected
during the third science run (S3) of the LIGO detectors, no detection of
gravitational waves was made and we set an upper limit on the rate of
coalescences of stellar mass binaries. The inspiral of stellar mass compact
objects into super massive black holes will radiate gravitational waves at
frequencies detectable by the planned space-based LISA mission. In Chapter 3 we
describe the development and testing of a computationally cheap method to
detect the loudest few extreme mass ratio inspiral events that LISA will be
sensitive to.
