TITLE: What should be taught in a 1st course on Riemann Surfaces?
QUESTION [20 upvotes]: I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term.  The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold theory, but may not have had much expose to algebraic geometry.  I will loosely follow the book  Introduction to Algebraic Curves by Griffiths.  In particular, I hope to spend a minimum amount of time developing basic machinery (e.g. sheaf theory) and to start doing concrete geometry (e.g. canonical models of curves of genus up to 4) as soon as possible.  
My question is: what are some good concrete, accessible  geometric topics in Riemann Surface/Curve theory that aren't in the standard textbooks?  
Let's say that the standard textbooks are the book I mentioned and those discussed:
  here.

REPLY [7 votes]: As Gerald said, really understanding some specific surfaces is useful.  And not just the compact ones!  Remember that Riemann's ideas were based on analytic continuation, not algebra.
And if you really mean Riemann surfaces, then Divisors and monodromy (compute some actual monodromy matrices!).
And if you want to cover material which is not in the standard textbooks, cover in depth the relation between Riemann surfaces and differential equations.