Why do some carmakers use belts and some use chains. I would think that chains would be superior-- they last longer. I think Audi uses timing belts rather than chains, is this correct? I'm not sure, but I think that Saab, BMW and Mercedes all use chains.

There is no answer to which is better, a belts and chains each have their advantages/dissadvantages. Timing chains do last longer, however they do stretch over time allowing the possibility a little 'slop' in valve timing, with a belt this is a non-issue, if the belt stretches it will break plain and simple. This is why manufacturers have replacements intervals for belts. Chains typically make more noise, now this is getting to be less of a problem with the advent of coposite chain covers that absorb noise. Chains also mean more reciprocating mass, this robs power a little bit. In addition, many manufacturers stay away from chains in high reving engines, because of the centrifugal force, you would have to have a complex set of guides. For the MB example-most of their engines are fairly slow turning (less than 6000RPM) and MB wants absolute long term durability. Generally most Japanese cars will have belts. It just comes down to which philosophies the manufacturer wants to go with. And unfortunately there are no 'etched in stone' standards of which is the better application. You can find some high-reving engines that use chains effectively too.