The concept of sets is as ancient as the concept of numbers. The modern set theory developed by George Cantor (1845-1918) is an essential notion in the whole of mathematics. All known mathematical concepts can be reduced to the ideas of set and membership in a set. The mathematical concepts explained with the aid of set theory, have an intrinsic clarity and exactness. Thus, set theory plays a significant part in bringing together different branches of mathematics. Even though accuracy is imperative for mathematics, we can observe vagueness in the definition of a set itself. Philosophers, logicians and mathematicians have spent quite a time to overcome this with an alternative notion. They came up with many notions, and the most significant among them come under Generalized sets, which comprises of fuzzy sets, rough sets, soft sets, multi sets, and many more. Category theory paves the way to describe and compare objects with similar and different properties. It has a pivotal role in modern mathematics and computer programming. The applications of category theory include everything from pure and applied mathematics, economics, philosophy, engineering and linguistics.