A database which uses a functional language as its query language.
Databases would seem to be an inappropriate application for functional languages since, a purely functional language would have to return a new copy of the entire database every time (part of) it was updated. To be practically scalable, the update mechanism must clearly be destructive rather than functional; however it is quite feasible for the query language to be purely functional so long as the database is considered as an argument.
One approach to the update problem would use a monad to encapsulate database access and ensure it was single threaded. Alternative approaches have been suggested by Trinder, who suggests non-destructive updating with shared data structures, and Sutton who uses a variant of a Phil Wadler's linear type system.
There are two main classes of functional database languages. The first is based upon Backus' FP language, of which FQL is probably the best known example. Adaplan is a more recent language which falls into this category.
More recently, people have been working on languages which are syntactically very similar to modern functional programming languages, but which also provide all of the features of a database language, e.g. bulk data structures which can be incrementally updated, type systems which can be incrementally updated, and all data persisting in a database. Examples are PFL [Poulovassilis&Small, VLDB-91], and Machiavelli [Ohori et al, ACM SIGMOD Conference, 1998].
Query optimisation is very important for database languages in general and the referential transparency of functional languages allows optimisations which would be harder to verify in presence of side-effects.
[Trinder, P., "Referentially transparent database languages", 1989 Glasgow Workshop on Functional programming]
[Breazu-Tannen et al., DBPL-91].