TITLE: What is a cohomology theory (seriously)?
QUESTION [73 upvotes]: This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"? 
I know that there exist generalized cohomology theories, Weil cohomology theories and perhaps one might include delta-functors, which describe (some of) the properties of explicit cohomology theories. But is there now a concept that underlies all these concepts? Or has the term "cohomology theory" been used so inflationary that the best thing one could say is that a cohomology theory is a sequence of functors into an "algebraic category" (whatever that is)?
Moreover, what is the difference between a cohomology theory and a homology theory? Of course I am aware of the examples like singular (co-)homology and I know the difference in this situation, but in general?

REPLY [8 votes]: I'd say a cohomology theory is a misnomer.  A theory really ought to be significant, make predictions, help us think about things, help us prove theorems.  At some point mathematicians decided to start giving away the word "theory" for free.    Newtonian mechanics, evolution, calculus -- those are theories.  
Is it a lack of imagination on our part?  It seems like anything that hasn't earned a proper name gets called "X theory" nowadays, for various values of X.  I'm glad differential geometry was invented in a previous era. Our contemporaries would have saddled the subject with some glorious name like "geometry theory" or "G-theory". 
(not exactly sure if tongue is in cheek)