Several approaches have been proposed for the solution of this problem but development is currently still very much in progress. Some of the more successful approaches are principal components analysis and independent components analysis.
One of the practical applications being researched in this area is medical imaging of the brain with magnetoencephalography. This kind of imaging involves careful measurements of magnetic fields outside the head which yields an accurate 3D-picture of the interior of the head. However, external sources of electromagnetic fields such as a wristwatch on the subjects arm, will significantly degrade the accuracy of the measurement. Applying source separation techniques on the measured signals can help removing undesired artifacts from the signal.
Another application is the separation of musical signals. For a stereo mix of relatively simple signals it is now possible to make a pretty accurate separation, although some artefacts remain.