[jesuslop]

Its of course the theory behind monads that since Eugenio Moggi are used to model computational effects in pure functional languages. Effects such as state, optional return types (used in turn for error handling) (maybe monad), input/output (reader writer monad) and others. Beyond effects, Wadler used monads for parsers (monadic parsing).

The Curry-Howard "isomorphism" (slogan: propositions are types, proofs are programs/functions) map code to logic in a categorical way described first by certain book of Lambek-Scott with uses in formal software verification.

Categories provide abstraction. You first distill the behavior of how Haskell (or you other pet functional language) work with Hask, the category of Haskell types, and then you can apply your abstract distillate to other categories and obtain task-oriented, tailored computing concepts that enrich bare language capabilities, providing applications including 1) probabilistic programs 2) automatic differentiation. Conal Elliott has very concrete work along this lines. When he speaks of CCCs (following Lambek) he alludes to cartesian closed categories, the crucial property of having a type constructor for function spaces and higher order functions. See his "compiling to categories" for a very concrete, hands-on feel. Another application he shows is in hardware synthesis (baking your functional algorithm to a netlist of logical gates for automating the design of custom hw accelerators).

In short, why categories? computational effects, formal verification and the equivalence of simply-typed lambda-calculus with cartesian closed categories, with lambda-calculus being the backbone of functional programming language semantics.

[Twey]

I'd phrase this a tiny bit differently: monads give a model of effects in _impure_ languages and are important for that reason. The fact that Haskell chooses to emphasize monads in the language itself is cool, but their utility is not restricted to pure functional languages; quite the opposite! In a pure functional language you don't need to think about effects at all, and the only model you need is mathematical functions, which are much simpler.