Every natural number greater than 1 is either prime or composite. Those numbers which have only two factors i.e., 1 and itself are considered as prime and rest are considered as composite numbers. The property of a number for being prime is called primality. In this survey paper we will deal with the algorithms that can check whether the number is prime or not i.e., primality test. This study is the detailed survey of probabilistic and deterministic algorithms like Fermat’s theorem of primality test, AKS theorem, Miller Rabin’s test, Solvay Strassen’s theorem etc. We will discuss different parameters regarding algorithms which are best for testing large prime numbers. Many aspects will be considered while discussing these algorithms.