Monads in Haskell can be thought of as composable computation descriptions. The essence of monad is thus separation of composition timeline from the composed computation's execution timeline, as well as the ability of computation to implicitly carry extra data, as pertaining to the computation itself, in addition to its one (hence the name) output, that it will produce when run (or queried, or called upon). This lends monads to supplementing pure calculations with features like I/O, common environment or state, and to preprocessing of computations (simplification, optimization etc.).

Thus in Haskell, though it is a purely-functional language, side effects that will be performed by a computation can be dealt with and combined purely at the monad's composition time. Monads thus resemble programs in a particular DSL. While programs may describe impure effects and actions outside Haskell, they can still be combined and processed ("assembled") purely, inside Haskell, creating a pure Haskell value - a computation action description that describes an impure calculation. That is how Monads in Haskell separate between the pure and the impure.

The computation doesn't have to be impure and can be pure itself as well. Then monads serve to provide the benefits of separation of concerns, and automatic creation of a computational "pipeline". Because they are very useful in practice but rather mind-twisting for the beginners, numerous tutorials that deal exclusively with monads were created (see monad tutorials).

Most common applications of monads include:

• Representing failure using Maybe monad
• Nondeterminism using List monad to represent carrying multiple values

Monads can be viewed as a standard programming interface to various data or control structures, which is captured by the Monad class. All common monads are members of it:

(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b
return :: a -> m a
fail :: String -> m a

In addition to implementing the class functions, all instances of Monad should obey the following equations, or Monad Laws:

return a >>= k  =  k a
m >>= return  =  m
m >>= (\x -> k x >>= h)  =  (m >>= k) >>= h

See this intuitive explanation of why they should obey the Monad laws. It basically says that monad's reactions should be associative under Kleisli composition, defined as (f >=> g) x = f x >>= g, with return its left and right identity element.

fmap ab ma = ma >>= (return . ab)

However, the Functor class is not a superclass of the Monad class. See Functor hierarchy proposal.

## Special notation

In order to improve the look of code that uses monads Haskell provides a special syntactic sugar called do-notation. For example, following expression:

thing1 >>= (\x -> func1 x >>= (\y -> thing2
>>= (\_ -> func2 y (\z -> return z))))

which can be written more clearly by breaking it into several lines and omitting parentheses:

thing1 >>= \x ->
func1 x >>= \y ->
thing2 >>= \_ ->
func2 y >>= \z ->
return z

can be also written using the do-notation as follows:

do
x <- thing1
y <- func1 x
thing2
z <- func2 y
return z

Code written using the do-notation is transformed by the compiler to ordinary expressions that use Monad class functions.

When using the do-notation and a monad like State or IO programs look very much like programs written in an imperative language as each line contains a statement that can change the simulated global state of the program and optionally binds a (local) variable that can be used by the statements later in the code block.

It is possible to intermix the do-notation with regular notation.

More on the do-notation can be found in a section of Monads as computation and in other tutorials.

Commutative monads are monads for which the order of actions makes no difference (they commute), that is when following code:

do
a <- actA
b <- actB
m a b

is the same as:

do
b <- actB
a <- actA
m a b

Examples of commutative include:

Monads are known for being deeply confusing to lots of people, so there are plenty of tutorials specifically related to monads. Each takes a different approach to Monads, and hopefully everyone will find something useful.

See the Monad tutorials timeline for a comprehensive list of monad tutorials.

An explanation of the basic Monad functions, with examples, can be found in the reference guide A tour of the Haskell Monad functions, by Henk-Jan van Tuyl.

Implementations of monads in other languages.

Unfinished:

And possibly there exist:

• Standard ML (via modules?)