Abstract: Music audio generation spans instrumental clips, full songs with vocals, edits of existing recordings, and systems that play alongside a musician in real time. This monograph explains the methods behind these systems for readers familiar with neural-network training who are new to music generation. It introduces neural audio codecs and autoencoders, autoregressive and masked token models, diffusion and flow matching, sampling acceleration, and preference optimization in a shared notation. Worked examples connect training objectives and sampling procedures to published systems. The application sections relate the musical requirements of instrumental generation, songs, editing, and live performance to model design. Comparison tables record representations, methods, inputs, tasks, and release information documented in papers and official sources. Symbolic composition, speech, and sound effects are included where they connect to music audio. We explain how to interpret automatic metrics and learned judges, design and report listening tests, and assess evidence about data provenance, memorization, attribution, and output detection. Selected legal decisions and disputes illustrate the implications for training and release. The reviewed systems combine established generative paradigms with new representations, planning stages, and post-training procedures. Differences in data, scale, and evaluation limit what cross-system comparisons can establish about the sources of reported gains.