The Einstein@Home project has been created to perform all-sky searches for previously unknown continuous gravitational-wave (CW) sources using data from the Laser Interferometer Gravitational-Wave Observatory (LIGO) detector instruments in Washington and Louisiana, USA. The primary class of target CW sources is rapidly rotating neutron stars (including pulsars) which are expected to emit gravitational waves due to a deviation from axisymmetry. Besides validating Einstein's theory of General Relativity, direct detection of gravitational waves would also constitute an important new astronomical tool. As most neutron stars are electromagnetically invisible, gravitational-wave observations might allow completely new populations of neutron stars to be revealed. A CW detection could potentially be extremely helpful in neutron-star astrophysics and would eventually provide unique insights into the nature of matter at high densities.