In this thesis, numerical methods of interpolation of complex-valued functions of a real argument and their application to the Runge function are discussed. The theoretical background for the following methods of interpolation of complex one-dimensional functions of a real argument is given: polynomial interpolation, Newton interpolation, Lagrange interpolation, trigonometric interpolation, Hermite cubic interpolation, rational interpolation and Chebyshev interpolation. Interpolation points can be equidistant, but much better interpolation of the function is obtained if the interpolation points are Chebyshev points of the first type, which do not include boundary points of the observed interval, or Chebyshev points of the second or third type that include boundary points of the observed interval. In addition to the points described in this paper, the zero-points of Legendre polynomials, also called Gauss-Legendre points, and the extreme points of Legendre polynomials, also called Gauss-Lobatto-Legendre points, are used as interpolation points. In numerical interpolation, larger or smaller oscillations occur in relation to the original function depending on which interpolation points are used and which interpolation method is used. In Lagrange interpolation, the oscillations increase if the degree of interpolation increases, which does not happen in polynomial interpolation, Newton interpolation, rational interpolation, and Chebyshev interpolation. In some numerical applications it is advisable to use Hermite cubic interpolation between interpolation points, whereas in some other numerical applications Chebyshev interpolation is imposed as the optimal solution.