Modules for associative or Lie algebras are just vector spaces acted on by the algebra. This means that we can take the product of an algebra element with a module element in a sensible way. Typically, one cannot take the product of two module elements, though. However, in certain circumstances, it is possible to make a module into an associative or Lie algebra, in a way that is compatible with the action. I will start with some examples for finite-dimensional Lie algebras, then infinite-dimensional Kac-Moody Lie algebras and finally quantized enveloping algebras. The products we get will turn out to involve braidings, giving us braided Lie algebras and braided enveloping algebras. I will also demonstrate one use for this extra structure, namely gluing together the module and the original algebra to get bigger algebras.