A green roof, also known as a living roof, is a roof of a building that is partially or entirely covered with vegetation (grasses or sedum, typically). Green roofs serve many purposes for a building, for example to absorb rainwater, to insulate the rooftop, creating habitats for wildlife and lowering urban air temperatures. In cities particularly, green roofing is also called 'urban greening' and is a very popular trend at the moment. 
A green roof can have several advantages, some of which are detailed below:
- Reducing heating (by adding mass and thermal resistance value)
- Reduce cooling (by evaporative cooling) loads on a building by 50-90%
- Reduce stormwater runoff
- Natural Habitat Creation
- Filter pollutants and carbon dioxide out of the air
- Filter pollutants and heavy metals out of rain water
- Help to insulate a building for sound
- Increase the lifespan of a roof drastically
- Increase real estate value
- Reduce energy usage

The main disadvantage of green roofs is the higher initial cost of the building structer, waterproofing systems and root barriers. There is also a degree of maintenance involved although the ongoing costs of most green roofs are relatively low.

Sources: here and here

Sedum on an extension

Urban Greening

Commercial urban sedum roof

Sedum roof