According to the Wikipedia article on the Abacus, Ettore Carruccio stated in Mathematics and Logic In History and In Contemporary Thought that the Old Babylonians:
may have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations
translated by Isabel Quigly
The Old Babylonians, however, performed more complex mathematics than basic arithmetic. In the well referenced Wikipedia article on Babylonian mathematics it is stated that there is evidence they solved equations algebraically. Analysis of the Plimpton 322 tablet even seems to indicate they calculated a list of Pythagorean triples "too many and too large to have been obtained by brute force".
What methods then did the Old Babylonians use for solving equations? Were other devices aside from the abacus employed, or were these mostly manual methods on tablets? How did their manual methods differ from those we use today?
I'm not sure how much information we have on their mathematics. If there is a lot available, then this may be a little broad--in that case a summary of the state of research with links to good starting places to read about the research would make a useful answer.
Referenced Wikipedia articles:
Note: I have discovered papers such as this, but I am unable to ascertain whether the paper is reputable (I'm unfamiliar with the scholars in this field) or whether it contains much in the way of an answer to my question. It would be nice to know that before purchasing access to a paper at random. In this case, the paper seems to be freely accessible and at a glance appears scholarly, yet I still cannot verify if it represents the current state of research.