The purpose of this work is to study how to communicate reliably through a noisy communication channel. In particular, we focus on the low-density parity-check error-correcting codes, which were introduced by Robert Gallager in his PhD thesis in 1963. Nowadays, in an very connected world, it can be said that the importance of this subject is fundamental. Indeed, we can find several situations where we have a transmission of information for example between cellular phones, the earth and a satellite and storage within the hard disk of a computer. Moreover, in last few years, with the development of chat-bots and with the purpose of creating good artificial neural networks, error-correcting codes has acquired greater importance, building a strong relationship between machine learning and information theory. This work is mostly based on the book “Information, Physics and Computation” by Marc Mézard and Andrea Montanari.