At the moment I'm aware of the following methods to integrate side-effects into purely functional programming languages:
Monads are often cited to be the most effective and most general way to do this.
Which other methods exist? How do they compare?
On
The very simplest method is to simply pass around the environment between the functions. This is often used to teach scheme.
On
To me a more general way is via a monad/comonad pair. This generalizes the common "monad" approach which should correctly be called the "strong monad" approach, since it only works with strong monads.
Moving to a monad/comonad pair allows effects to be modeled that result in some variables no longer being available. An example where this is useful is the effect of migrating a thread to another host in a distributed setting.
An additional method of historical interest is to make the whole program a function mapping a stream/list of input events to a stream/list of output events. See: "How to Declare an Imperative" by Phil Wadler: http://www.cs.bell-labs.com/~wadler/topics/monads.html#monadsdeclare
Arrows, which are more general than monads.