This Chapter introduces techniques for approximating functions using polynomials, both at a point and over an interval, which are foundational for understanding several advanced quantum algorithms. The Taylor series expansion for approximating an analytical function around a point is presented with its error bounds around the approximation point. Chebyshev polynomials are presented as a route to approximating analytic functions with uniformly bounded error over an interval. Subsequently, evaluation of analytical functions of matrices is presented with the special case of diagonalizable matrices. Finally, we iscuss the matrix exponentiation problem—an elementary task in quantum computation—along with error bounds for approximate exponentiation using the Baker–Campbell–Hausdorff formula.