Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed-matter physics. In this paper basic properties and recent developments of Chebyshev expansion based algorithms and the kernel polynomial method are reviewed. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems are discussed in detail. In addition, an illustration on how the kernel polynomial method is successfully embedded into other numerical techniques, such as cluster perturbation theory or Monte Carlo simulation, is provided.