Or does the wind just blow off the rain when they go really fast?
Yes most large transport aircraft have variable speed windshield wipers but they are normally used in fairly heavy rain, predominantly during landing and occasionally during takeoff. There were some aircraft that used high pressure air blowing across the windshield to keep rain off but most use a wiper system. Each side is independent from the other side.They are never used in cruise conditions. At cruise speed airflow keeps the forward windshields clear. Light airplanes generally don't have them.
Wow, some really interesting, and rather amazing, answers so far. Actually, the majority of modern airplanes do not have windshield wipers, and almost no light planes (single and twin engine) have them. Most aircraft use plexiglas for windshields, and one pass with a wiper blade would scratch and craze it to the point of being unusable. In flight, airflow clears water from the windshield quite effectively, and, when in the clouds, flight is by reference to instruments inside the cockpit. On breaking out of the clouds, the high speed air flow helps keep the windshield clear enough of rain for good visual acuity. Almost all have defrosters, which blow heated air on the inside of the windscreen, but that isn't really effective if you get into icing conditions - a heated windshield, or at least an alcohol windshield spraybar, is the only defense against icing over the windshield in that area. A number of large transport category airplanes do have wipers, primarily for ground operations, they are generally not used in flight.
Many airplanes do have windshield wipers. Some of the smaller planes and those that were in service during WW1 do not have winshield wipers. Airplanes fly through clouds of dust and water. They also need to fly though any kind of weather when they land so they very much need their windshield wipers.
