In this review, biometric systems are studied from an information theoretical point of view. In the first part biometric authentication systems are studied. The objective of these systems is, observing correlated enrollment and authentication biometric sequences, to generate or convey as large as possible secret keys by interchanging a public message, while minimizing privacy leakage. Here privacy leakage is defined as the amount of information that this public message contains about the biometric enrollment sequence. In this setting also the secrecy leakage, that is, the amount of information the public message leaks about the secret key, should be negligible. Next identification biometric systems are investigated. These systems should be able to identify as many individuals as possible while being able to assign as large as possible secret keys to each individual and again minimize the privacy leakage. To realize these systems public reference data are stored in the database. Leakage is defined with respect to these reference data. For all these biometric systems fundamental limits are determined in the current work. Finally, a popular practical construction for biometric systems, fuzzy commitment, is analyzed with respect to both its theoretical performance and performance related to the code choice.