
  "Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
  
  Felix Klein


What notions are used but not clearly defined in modern mathematics?



To clarify further what is the purpose of the question following is another quote by M. Emerton: 


  "It is worth drawing out the idea that even in contemporary mathematics there are notions which (so far) escape rigorous definition, but which nevertheless have substantial mathematical content, and allow people to make computations and draw conclusions that are otherwise out of reach."


The question is about examples for such notions.

The question was asked by Kakaz

One of the most important contemporary mathematical concepts without a rigorous definition is 
quantum field theory (and related concepts, such as Feynman path integrals).  

Note: As noted in the comments below, there is a branch of pure mathematics --- constructive field theory --- devoted to making rigorous sense of this problem via analytic methods.  I should add that there is also a lot of research devoted to understanding various aspects of field theory via (higher) categorical points of view.  But (as far as I understand), there remain important and interesting computations that physicists can make using quantum field theoretic methods which can't yet be put on a rigorous mathematical basis.
