South Wales Police saw the biggest increase, with 38 separate incidents over the past year - up from only four during the previous three years.
Gwent Police had 19 complaints in 2015-16 and Dyfed-Powys had nine. North Wales Police had just one since 2010.
Drones were also reported over Swansea Prison twice last year.
A Freedom of Information (FOI) request revealed both the drone and a mobile phone were seized in the first incident, although no suspects were traced on either occasion.
Anyone found using drones in an attempt to get contraband into prisons could be punished with a jail sentence of up to two years.
Meanwhile, a second FOI request revealed the increase in calls to police in Wales about drones had increased in 2015-16.
They showed:
Drones are becoming increasingly accessible, with advances in technology leading to many people being able to afford a small device, often with a camera attached. Some basic models now cost less than Â£100, with more advanced types commonly available for Â£700-Â£1,000.
The CAA and police around the UK have successfully carried out four prosecutions for drone misuse, all resulting in fines.
The use of drones is governed by the law, and the CAA has a set of rules called the "drone code".
For instance, they must not be flown higher than 400ft (about 122 metres), users must be able to see them at all times, and drones with cameras cannot be flown within 50 metres (164ft) of people, vehicles, buildings or structures.
Police are dealing with a significant rise in calls about drones amid the growing boom in "cameras in the sky", figures released to BBC Wales show.