``Cut and Paste'' arguments are a common tool used to understand
fundamental groups of curves over the complex numbers. I will discuss
such arguments in the classical setting and then explain their
analogues in positive characteristic. Cutting and pasting includes
simple techniques over Zariski patches, and more sophisticated tools
such as Grothendieck's Existence Theorem and Formal Patching.
I hope to mention several recent, and not so recent, results which have
been proved by cutting and pasting. For example, the proof (by
Harbater and Raynaud) of the Abhyankar Conjecture relied in large part
on such techniques.