It is generally believed that string theory admits a huge number of vacua with a range of different vacuum energies. Chain inflation takes place in this string theory "landscape" as the universe tunnels rapidly through a series of ever lower energy vacua, with an early period of rapid tunneling followed by a later period of slow tunneling, such as may be required to explain today's dark energy and small cosmological constant. Each tunneling event provides a fraction of an e-folding of inflation, so that hundreds of events are required to provide the requisite amount of inflation.