We study the problem of polynomial interpolation. Its solution with the Vandermonde matrix, and with a Lagrange nodal basis are then presented, and error estimates are provided. The Runge phenomenon is then illustrated. Hermite interpolation then is studied, its solution is given, and error estimates are provided. The problem of Lagrange interpolation is then generalized to the case of holomorphic functions on the complex plane, and error estimates are provided. A more efficient construction, via divided differences, is then given for the interpolating polynomial. We extend the notion of divided differences, in order to use them to provide error estimates for polynomial interpolation.