In the fields of signal processing and music information retrieval (MIR), the task of decomposing a music recording into musically meaningful sound sources is referred to as source separation. As an example, a funk instrumental may consist of drums, bass, and saxophone playing together. Given this mixture, the goal is to recover the individual instruments' sounds|as if they had been recorded in isolation. In this thesis, we focus on separating and restoring drum sounds from music recordings. Drums typically emphasize and shape the rhythm, and often define the musical style. In contrast to melodic instruments, drums mainly produce percussive and inharmonic sounds that may overlap considerably in time and frequency. The challenge is to extract perceptually convincing source signals without introducing audible artifacts. We systematically approach this research problem in three parts of the thesis, each with a different perspective. The first part is about automatic drum transcription (ADT), i. e., the detection and classification of drum sound events in music recordings. Based on a literature review, we give a comprehensive account of prior work on this topic. We then select two families of state-of-the-art ADT algorithms based on Non-Negative Matrix Factorization (NMF) and Recurrent Neural Networks (RNN), and compare their performance in a controlled experimental setting. In the second part of this thesis, we focus on drum sound separation. Aiming for a better understanding of the capabilities and limitations of NMF, we investigate how to embed side information and how to exploit drum-specific properties. As one main contribution, we introduce suitable constraints that help steer iterative NMF methods towards a meaningful solution. Furthermore, to improve the perceptual quality of the separated drum sounds, we propose dictionary-based restoration schemes for repairing cross-talk artifacts. Finally, we investigate signal reconstruction methods and develop a transient restoration technique that is suited to sharpen ...