The Discrete Logarithm Problem (DLP) is one of the most used mathematical problems in asymmetric cryptography design, the other one being the integer factorization. It is intrinsically related to the Diffie-Hellman problem (DHP). DLP can be stated in various groups. It must be hard in well-chosen groups, so that secure-enough cryptosystems can be built. In this chapter, we present the DLP, the various cryptographic problems based on it, the commonly used groups, and the major algorithms available at the moment to compute discrete logarithms in such groups. We also list the groups that must be avoided for security reasons. Our computational model will be that of classical computers. It is to be noted that in the quantum model, DLP can be solved in polynomial time for cyclic groups.