"Old camera connected to new flash" shouldn't be a problem, provided that you have the appropriate connector.
In some cases you need to be careful with the additional connection points on hot shoes and hot shoe connections.
You will, of course, only get to benefit from the additional functions offered by "dedicated" flashes if your camera communicates with them appropriately.
There can be problems with the "new camera with old flash" scenario, if the electronics on the flash present too high a trigger voltage for the switching circuit on the camera. Most new cameras synch flashes electronically, while older cameras have mechanical synch circuitry that handles higher trigger voltages.
This site includes a survey of synch trigger voltage tests: http://www.botzilla.com/photo/strobeVolts.html
EDIT: Wein used to make "Safe Synch" connectors to help isolate the high voltages. Paramount Cords offer custom varieties.