Polynomial interpolation is the process of determining the polynomial of the lowest possible degree whose graph passes through given nodes. In this final work, three ways of computing an interpolation polynomial are explained: using the Lagrange polynomial interpolation form, using the Newton polynomial interpolation form and directly, by Gaussian elimination method on the Vandermond matrix. In addition to the methods explained, two important interpolation theorems are listed, along with their proofs. These are the theorem that claims that the interpolation polynomial is unique and the theorem that gives us a formula for calculating the error estimate when approximating a function by an interpolation polynomial. The disadvantage of polynomial interpolation is that the calculation of such a polynomial, and its evaluation, is very time consuming for a large number of nodes. Therefore, such interpolation is rarely used in practice if the number of nodes to be interpolated is large. In such cases, a better solution is polynomial interpolation by parts, the so-called splines. Also, when interpolating some functions, the Runge phenomenon can occur, when a sequence of interpolation polynomials does not converge (pointwise) to a function that we approximate when the number of nodes increases.