Spectral decomposition by nonnegative matrix factorisation (NMF) has become state-of-the-art practice in many audio signal processing tasks, such as source separation, enhancement or transcription. This chapter reviews the fundamentals of NMF-based audio decomposition, in unsupervised and informed settings. We formulate NMF as an optimisation problem and discuss the choice of the measure of fit. We present the standard majorisation-minimisation strategy to address optimisation for NMF with the common -divergence, a family of measures of fit that takes the quadratic cost, the generalised Kullback-Leibler divergence and the Itakura-Saito divergence as special cases. We discuss the reconstruction of time-domain components from the spectral factorisation and present common variants of NMF-based spectral decomposition: supervised and informed settings, regularised versions, temporal models.