I've had propellers run away in piston and turopropeller airplanes. On the piston side, I've had it happen on small single engine horizontally opposed installtions with two blade constant speed props, and large radial three and four blade ham standard hydromatic props.
How can you tell you're in overspeed? Look at your tachometer. Is it indicating faster than your upper limits? You're overspeeding. Is speed increasing above where you set it, or above were you saw it last and you haven't touched the prop control? It's overspeeding.
In a nutshell, an overspeed is caused by a governor malfunction in which the governor allows the propeller to go to the fine pitch/high RPM position. Your propeller has just become a fixed pitch propeller at the low pitch stops. It's going to go as fast as either the engine will drive it, or the slipstream will drive it...which ever imparts more energy to the prop. An overspeed may also be caused by other problems, such as a failure of a low pitch stop, or a runaway engine, or excess airspeed.
If your propeller is running away, engine speed increasing, and you can't bring it back, you need to get it under control, and your only choice may be to shut it down. However, you can try other things first. Reducing power, right to idle, if necessary, may fix your problem. If you reduce power and your RPM falls within limits, and you can keep it there at your current airspeed, then you can use that amount of power...whatever it is, for the rest of the trip in level flight. Any more at that airspeed is going to overspeed the prop.
The secret is slowing down. Pull the power back, get to a slower airspeed where the slipstream isn't driving the propeller, and then start applyng power again. Let the engine impart torque to the propeller and the propeller convert that to thrust, rather than letting it absorb slipstream energy as the slipstream drives it. This is most important in a multi engine airplane, where a propeller that's overspeeding and aborbing energy will be similar to one going into beta or reverse...it's not doing ou any good. It's windmilling out there, if you've got the power back and it's still trying overspeed; the slipstream is driving it, and not your engine. It's aborbing energy from the slipstream, creating a lot of drag on that side, and needs to be either feathered or slowed down.
Many moons ago we lost a C-119. The airplane went into overspeed, and the crew pulled the power back. The aircraft began descending with the reduced power, but it also slowed down, and as it did, the RPM came back under control. The crew applied power now that the airspeed was under control, the aircraft climbed, the propeller ran away again, and they reduced power. Over the course of about 45 minutes, this cycle continued, grdually losing altitude, until it crashed on the high desert floor.
If the crew would have just kept it slow and used engine power and a slower airspeed, they'd have been fine. Slow down, don't let the slipstream drive the propeller, and let the propeller take your engine power and give you something meaningful in return instead of working against you.
Watch your tach, follow your AFM proceedures.