TITLE: What is the role of fixed point theorems in modern mathematics?
QUESTION [5 upvotes]: About Fixed Point Theorems, Wikipedia says: 

Results of this kind are amongst the most generally useful in
  mathematics.

This seems an accurate statement: indeed, there are many journals dedicated exclusively to this topic and many monographs have been published. 

Could you elaborate on why this kind of theorem is important to the development of current mathematical research? In particular, what is the relationship between fixed point theory and analysis?
Also, could you recommend some self-contained but comprehensive monographs?

REPLY [2 votes]: Many problems can be turned into 'fixed point theorem' problems.
One of the most useful classes of such problems relates to existence/uniqueness theorems for differential equations.