In Ken Follett's series of books about the fictional town of Kingsbridge, the monastery essentially acts as lord over the surrounding area. They collect rent on lands they own, hear cases between the town's residents, and perform other duties that the local lord would. Did this actually happen in Europe?
Yes, monasteries in the medieval period were in fact powerful lords of feudal estates.
Most bishops and monasteries had large landholdings, which had been granted to them over the years by kings, or great lords. At one stage, the church was the largest landholder in Europe.
The monastery at Luxeuil in Burgundy had enough land to support 15,000 manors. Another bishop of St Martin of Tours in France ruled something like 20,000 serfs.
As with any other feudal landlord, the tenants of the monastery would owe rents and duties to their landlord (in addition to the tithes they owed to the church).