Symmetric polynomials arise in many different fields, such as combinatorics, representation theory, analysis, algebraic geometry, topology, probability, mathematical physics. They are a very powerful tool for computation because a lot of information can be encoded in simple expressions. In this lecture I will construct different families of symmetric polynomials and show how to do simple manipulations with them. Then I will try to demonstrate the power of symmetric polynomials in some applications: if time permits, I will compute some integrals, define zeta function of a dynamical system and compute some determinants.