In addition:

It was true up until the advent of advanced ECUs that higher octane gas alone wouldn't get you better performance - just that lower octane gas would damage your engine.

Most of today's cars, though, have knock sensors that allow the ECU to scale back performance for lower octane gas will still performing better on high octane gas. Most N/A cars are designed for the highest readily available octane of gas but won't completely break if a lower octane is put in (I would NEVER recommend doing this, though, at best your car will run like a dog).

Things get more interesting with forced-induction cars, though. Many FI cars these days are able to cope with higher boost than is possible with premium gas (due to preignition), and are tuned to allow this from the factory. Thanks to computer-controlled wastegates and supercharger clutches, they're now able to run scaled back on 91/93 octane but keep the ability to run higher boost if the gas allows. So many factory-stock F/I cars will automatically increase pressure if 100/103 octane gas is put in, sometimes leading to damn impressive results. Similar results are seen with E85 - if the fuel system on an FI car is able to handle E85 (which is 105+ octane as pure ETOH is 127-129 octane in the American system) the car can be tuned to handle incredible amounts of boost.