This Chapter describes numerical integration for approximate computation of the values of definite integrals, as well as techniques for differentiation of functions in the presence of noise. Integration over finite intervals is introduced via the classic trapezoidal, Simpson’s and Romberg’s methods, followed by a presentation of Gaussian, Clenshaw–Curtis and Gauss–Kronrod quadrature, as well as contour integration. The trapezoidal method and Gaussian quadrature are discussed again in the context of integration over semi-infinite and infinite domains, where we also present double-exponential methods based on variable transformation. Computation of Cauchy principal values is discussed, and several methods for the integration of rapidly oscillating functions are illustrated. Techniques of stable numerical differentiation as a notoriously ill-posed problem are described, based on three characteristic approaches from the theory of inverse problems: regularization of finite differences, which yields low-noise Lanczos and related families of noise-robust differentiators; differentiation by using smoothing kernels; and variational regularization of the derivative, which can be translated into the problem of solving an integral equation for the derivative function.