In this paper, relevant research advances in recent years on Low-density parity check codes are studied starting from its formulation by Gallager and rediscovery of these codes by McKay and Neal. Gallager construction of LDPC codes is described and their probabilistic decoding methods. Also, the construction by McKay is studied and the superposition method of construction is studied. Encoding and decoding techniques related to capacity-approaching error-correction technology is also studied. LDPC codes can also be described using a Tanner graph. Matrix representation and graphical representation of these codes are described. LDPC codes show better performance for more number of iterations for message passing decoding algorithm.