In this series of three lectures, we will discuss two important and relatively new methods in combinatorics. Firstly, the probabilistic method developed by Erdős and which has now seen numerous applications in various areas of mathematics such as number theory, linear algebra, additive combinatorics, real analysis, as well as in computer science. Secondly, we will give an introduction to the polynomial method and present few surprising applications of linear algebra type-arguments to problems in combinatorics and geometry.