Music recordings contain the mixed contribution of multiple overlapping instruments. In order to better understand the music, it would be beneficial to understand each instrument independently. This thesis focuses on separating the individual instrument recordings within a song. In particular, we propose novel algorithms for separating instrument recordings given only their mixture. When the number of source signals does not exceed the number of mixture signals, we focus on a subclass of source separation algorithms based on joint diagonalization. Each approach leverages a different form of source structure. We introduce repetitive structure as an alternative that leverages unique repetition patterns in music and compare its performance against the other techniques. When the number of source signals exceeds the number of mixtures (i.e. the underdetermined problem), we focus on spectrogram factorization techniques for source separation. We extend single-channel techniques to utilize the additional spatial information in multichannel recordings, and use phase information to improve the estimation of the underlying components.