How are buffer overflows used to exploit computers?

Question

How are buffer overflows used to exploit computers?

How is one able to execute arbitrary code simply by causing stack or heap overflows?

I understand that portions of the programs memory are overwritten that aren't supposed to be, but I don't see how this leads to one executing their own code. Also, must the 3rd party's malicious code be written in the target processors assembly language?

Answer 1

This is the most widely known document on the subject: Smashing the Stack for Fun and Profit

However, 'stack overflows' have nothing to do with buffer overflows. Stack overflows are generally just an error case in bad code that can't be exploited for anything outside of a crash (DoS).

Edit: You also asked about heap overflows. This is a good document on the subject: http://www.w00w00.org/files/articles/heaptut.txt