Fiats portfolio would grow considerably in 1958, and offer a large span of horsepower output, with the introduction of the 200, 300, 400 and 500 series tractors.
 
The 500 series came from OM's lineup of tractors, which was slowly but surely being integrated into the Fiat brand.
The tractor was originally named the OM 50, but was renamed the 512R, to fit with the new model designations Fiat used for its newest tractors. The tractor was sold under the OM and Fiat brand from 1959 till 1964. 

The OM engine hadn't been designed with a stressed crankcase in mind, and a strong subframe design was therefor used instead, to connect the front axle to the gearbox/differential casing, and carry the engine.
But like the 400 series, the 500 series did have front and rear width adjustment.

The gearbox was a seven forward and two reverse, and had PTO and locking differential on the rear axle as standard.
Some previous Fiat tractors had seen the use of a reduction gear. But on the 513R, Fiat and OM introduced a new mechanically activated on-the-go reduction gear named "Amplicouple", as an option for the 513R.
With results similar to later and more advanced hydraulically and electrically activated powershift systems we know today, the Amplicouple system (also known as "Automatic") would act as a quick engage and disengagement of the reduction gear in any gear, preventing the driver from having to come to a stop to engage the reduction gear, when momentum is most needed.
The reduction gear was "armed" by a leaver to the left of the driver. Power was sent through the PTO shaft, and could then be returned to the gearbox through the reduction gear on command via the clutch pedal.
This reduction gear was only ment for emergencies though, as its engagement would wear out clutch and throwout bearings prematurely. This coupled with the extra cost, ment that most costumers didn't order the Amplicouple reduction gear system.

Another innovative feature of this tractor, was the use of disc brakes. A very advanced technology for the time, having only a few years earlier been introduced to the public by Jaguar, on their Le Mans winning C-type racecar in 1953, equipped with Dunlops patented disc brake system.

Fiats new three-point hitch system named "Control-Matic" was introduced on the 300, 400 and 500 series, and was inspired by the revolutionary three point hitch system with draft control, developed and patented by Harry Ferguson in 1926.

The 512R used a OM developed four cylinder, water-cooled, diesel engine of 4.156 cc, capable of 50 hp at 1.600 rpm.
The 513R from 1960 would use an upgraded version of the 512R's engine, now delivering 55 hp at 1.800 rpm, and 240 nm at 1.200 rpm.
The last upgrade happened with the 513R from 1963, getting a new and larger OM developed four cylinder, water-cooled, diesel engine of 4.397 cc, now capable of 60 hp at 1.800 rpm, and 261 nm at as low as 1.150 rpm.

The Fiat 500 series lasted till 1964, where it became part of Fiats new and streamlined "Diamond-series" of tractor models, and was renamed the 615. Its platform would lay the foundation for the even larger 715 introduced one year later, finally replacing Fiats big but fundamentally old 80R tractor.