In this thesis, we are interested in Montgomery's pair correlation conjecture which is about the distribution of.the spacings between consecutive zeros of the Riemann Zeta function. Our goal is to explain and study Montgomery's pair correlation conjecture and discuss its connection with the random matrix theory. In Chapter One, we will explain how to define the Ftiemann Zeta function by using the analytic continuation. After this, several classical properties of the Ftiemann Zeta function will be discussed. In Chapter Two, We will explain the proof of Montgomery's main result and discuss the pair correlation conjecture in detail. The main result about the pair correlation functions of the eigenvalues of random matrices will also be proved. These two pair correlation functions turn out amazingly to be the same. Thus the full importance of Montgomery's conjecture is established.