Did WW2-era aircraft have constant-speed or variable-pitch props? This is slightly related to my other question on variable-pitch controls.

I want to know if WW2-era aircraft typically had variable-pitch or constant-speed propellers. Variable-pitch lets the pilot adjust the propeller blade pitch. "Constant speed" is the same as variable-pitch, but the engine does it automatically to keep the best performance at various airspeeds. This also lets the engine keep a constant RPM so it can always run at its optimal RPM, also increasing efficiency.

Anyway, were such things seen in WW2? Were they the norm, or the exception? Did single-engine aircraft have them, or were they more common on multi-engines?

I'm interested in all the nations who made aircraft, especially USA, UK, USSR, Germany, and Japan. And I'm interested specifically in the combat aircraft, like fighters and bombers, not necessarily the civilian transports of the era. This will give us a good idea if the extra complexity and weight was acceptable for combat aircraft of the era.
Variable pitch (with feathering) was virtually universal on (at least American) multi-engine fighters, bombers, and transport aircraft even early in the war: the DC-2 had feathering variable pitch, and it entered service in 1934.  The P-38 had feathering and variable pitch, and was in service before the US entered the war.

Constant speed propellers were also found on many single engine combat aircraft -- one was the Hellcat, which entered service in 1943, though it's very possible the technology was introduced earlier.  The P-47 Thunderbolt also had this feature.

When you're talking about aircraft that weigh several tons and need to deliver thousands of horsepower through a propeller, especially for combat operations, the weight and cost of variable or constant-speed propellers became negligible by comparison to the cost of the aircraft or the cost of fielding a losing design.