This is a chapter of the forthcoming Handbook of Linear Algebra, 2nd Edition (ed. L. Hogben). Spectral sets and K-spectral sets, introduced by John von Neumann in [vNe51], offer a possibility to estimate the norm of functions of matrices in terms of the sup-norm of the function. Examples of such spectral sets include the numerical range or the pseudospectrum of a matrix, discussed in Chapters 16 and 18. Estimating the norm of functions of matrices is an essential task in numerous fields of pure and applied mathematics, such as (numerical) linear algebra [Gre97, Hig08], functional analysis [Pau02], and numerical analysis. More specific examples include probability [DD99], semi-groups and existence results for operator-valued differential equations, the study of numerical schemes for the time discretization of evolution equations [Cro08], or the convergence rate of GMRES (Section 41.7). The notion of spectral sets involves many deep connections between linear algebra, operator theory, approximation theory, and complex analysis.