The aim of the project is to study constructible and transcendental numbers. The project starts with an overview of symmetric polynomials and field extensions. This is followed by a discussion of constructible numbers and their properties. In addition, it is proved that these numbers are algebraic which means that they are zeros for some polynomials over rational numbers. Next we discuss transcendental numbers and their history. Firstly we study the existence of transcendental numbers as shown by Liouville and Cantor respectively. Then we look at the irrationality of some real numbers and the transcendency of and e as shown by Lindemann and Hermite respectively. The project is concluded with a discussion of algebraic independence, transcendental extensions and transcendental bases.