Approximation theory is the mathematical study of how given quantities can be approximated by other (usually simpler) ones under appropriate conditions and the size and properties of the error introduced by approximation. The aim of this project is to study the approximation of bounded linear operator by compact operators T:X→Y. If X and Y are two normed linear spaces, then L(X, Y) is the normed linear space of all linear bounded operator from X to Y, and K(X, Y) the normed linear space of all compact linear operators. The project studies the proximinality of K(X, Y) in L(X, Y) in general, studies its' proximinal properties and gives examples for some none-proximinal cases. After a revision of the basic concepts on functional analysis in chapter 1 and 2, we explored the compact operators in the set of bounded linear operators in chapter 3. Additionally, the study provides an introduction to approximation theory and some properties of proximinal sets in chapter 4. Finally, we studied the approximation by compact operators between some classical Banach spaces.