Anybody living near the coast should be aware of tsunamis and ready to move to higher ground immediately after an earthquake or when tsunami warnings are in effect. Most tsunamis are too small to be noticed but the Tsunami Warning Center in Hawaii tracks them and sends out warnings when there is a chance of a large one. Unfortunately, some of the worst tsunamis occur near the epicenter of the originating earthquake and therefore give little warning. This was true of the devastating 1998 New Guinea tsunami, which was caused by an earthquake just off the coast. Tsunamis contributed to significant damage in the 1755 Lisbon quake and the 1964 Alaska quake , among others.
Page is a good source of more information.