Horses were the primary means of transportation for quite a while. Horses haven't disappeared. They're just used in different ways now. I imagine that's the future of oil-powered cars, too. Collectors, racers, those without access to grid power, or where power density and weight are mission critical (aircraft, for example).
Robin
Yeah, aircraft are a challenge.
They're already going more electric (the 787 made a number of interesting improvements in based on removing the compressed air/bleed air systems and replacing them with high voltage electricity and has a ~2.2kWh Lithium Ion battery which is mostly used to spin up the APU without ground power.) The new systems are lighter, more reliable, and actually improve cruise fuel efficiency by a couple percent through reduced engine losses.
Commercial aircraft will likely shortly be hybridized for ground operations - a couple companies are developing wheel-motors for the nose gear so the aircraft can push back and taxi on their own electrically without spinning up the main engines. I wouldn't be surprised if once that happens they start carrying somewhat larger batteries so they can put off cranking the APU longer - possibly even becoming something of a PHEV for ground operations: land, shut down all engines, taxi on battery power to the gate, plug in and charge the battery while passengers board, taxi out on battery and crank the main turbines just short of the runway. If the battery gets large enough, now you can do away with the APU entirely.
If solar panels get a little bit lighter and more efficient, I could see the aircraft starting to carry some (integrated into either the upper wing skin or the upper cabin structure) to further reduce the electrical load on the engines (Boeing is using high voltage AC generators for aircraft power, which means in principle they could probably be reversed to deliver some solar surplus to the high bypass turbofans to provide thrust, but it'd take major technological breakthroughs in both solar power and aerodynamics to make a jetliner fly mostly on Solar.) This could also allow the aircraft to use solar power for cabin climate control on the ground and taxiing.
However, it'll probably be a while before they go away from the main turbines - right now there's nothing on the horizon that can deliver that kind of power and range in anything like the same weight - short of some sort of beamed power system, at least (which has lots of problems, of course.)
If we get serious about global warming, the most likely solution for aircraft for the next 20-50 years will be some sort of carbon neutral liquid fuel. (The US Military has been exploring variations of Biodiesel for aircraft use with fairly good success - it's main challenge is staying fluid at stratospheric temperatures.)