Computers are nowadays much better than human mathematicians at calculations - they can multiply very large numbers together in a fraction of a second. But researchers in pure mathematics prove theorems. The theorem that there are infinitely many prime numbers cannot be proved by calculating more and more prime numbers - it needs a different approach, via logic and reasoning. It is not difficult to teach a computer the rules of logic and the axioms of mathematics. But then is it possible to go from these axioms, via many centuries of theorems, right up to teaching a computer some modern research mathematics? Recent evidence shows that this is now becoming feasible. Will computers soon be able to help humans prove theorems, or even prove new theorems by themselves? This we do not know. I will give an overview of where things are right now. The talk will be suitable for a general mathematical audience - no expertise in research mathematics or computer theorem provers will be assumed!