Ramsey Theory is named after Frank Plumpton Ramsey a young man wasrnespecially interested in logical and philosophy. Ramsey died at the age of 26rnin 1930 the same year that his paper on a problem of formal logic was pub-rnlished. His paper catalyzed the development of the mathematics _eld nowrnknown as Ramsey Theory. Problem in Ramsey Theory typically ask a ques-rntion of the form: How many people are required at gathering so that therernmust exist either three mutual acquaintances or three mutual strangers? Werncan rephrase this question as a problem in Ramsey Theory: How many ver-rntices do you need in edge 2 colored complete graphs for it to be necessaryrnthat there be either a red K3 (people who know each other) or a blue K3rn(people who do not know each other)? While many results in the subjects arernpublished each year, there are many questions whose answer remains elusive.rnRamsey Theory has played an important role in a plethora of mathematicsrndevelopment throughout the last century. Ramsey Theory is concerned withrnthe preservation of structure under partition it is the study of unavoidablernregularity in large structures. In this project, I explore some of the core ideasrnunderpinning Ramsey Theory and present a variety of problem to which itrncan provide interesting and elegant solution. Also we have see, the Ramseyrnnumber R(k; l) is the smallest natural number n such that in any red andrnblue coloring of the edges of the complete graph on n vertices, we are guar-rnanteed to _nd either a red Kk or a blue Kl. Furthermore, we have discussedrna multi-color example R(3; 3; 3) = 17, generalization of Ramsey Theoremrnand in_nite Ramsey Theory. Some Known Ramsey Theorem for bound onrnclassical Ramsey numbers are included.