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, the author presents 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. The authors also list the groups that must be avoided for security reasons. As a consequence, any pairing-friendly curve defined over a finite field of small characteristic should definitely be avoided. The pairing target group for any algebraic curve defined over a finite field is a subgroup in an extension of that finite field.