The International Monitoring System
In 1968, the world's nuclear powers negotiated an agreement called the Treaty on the Non-Proliferation of Nuclear Weapons, also known as the Non-Proliferation Treaty (NPT). That agreement sought to stop the testing of nuclear weapons. In 1997, the Comprehensive Test Ban Treaty Organization was formed to established a monitoring system that would detect unauthorized nuclear testing anywhere in the world and notify member nations.
Inspiration for this monitoring is partly due to an incident where a nuclear weapon was tested in the atmosphere and no nation claimed responsibility.
That network is operating today and has been since the mid-2000s. You wouldn't know it based on the political rhetoric from politicians who were vehemently against the Iran Nuclear Deal. To hear them tell it, we couldn't trust the Iranians to honor the deal in spite of the global network of monitoring stations and the constant video surveillance of their facilities.
This presentation details the monitoring network as well as other monitoring capabilities from space-based observation.