This is really two questions, but they are so similar, and to keep it simple, I figured I'd just roll them together:

Firstly: Given an established python project, what are some decent ways to speed it up beyond just plain in-code optimization?
Secondly:  When writing a program from scratch in python, what are some good ways to greatly improve performance?

For the first question, imagine you are handed a decently written project and you need to improve performance, but you can't seem to get much of a gain through refactoring/optimization.  What would you do to speed it up in this case short of rewriting it in something like C?
Run your app through the Python profiler.
Find a serious bottleneck.
Rewrite that bottleneck in C.
Repeat.
Speeding Up Python