Concerted observational and modelling programmes are underway to determine the mass balance of the Greenland Ice Sheet, and therefore help predict its response to future climatic change. We present results of meteorological modelling based on ERA-40 reanalysis data from the European Centre for Medium Range Weather Forecasts (ECMWF). Our novel surface-mass-balance history of the ice sheet for 1958-2003, is based on accumulation (snowfall minus evaporation/sublimation) modelling and a new monthly melt-water runoff model by Janssens & Huybrechts (Huybrechts 2002). These techniques combined yield valuable insights into the past and present state and variability of the Greenland ice mass and links with climate. Aspects of the validation of the new accumulation, runoff and SMB series are discussed. There was considerable interannual variability in snow accumulation, runoff and mass balance over the last 46 years. By comparing with long-term temperature, precipitation and accumulation records from the meteorological stations and ice cores, we discuss possible climatic factors forcing the ice in this period. There are distinct signals in runoff and SMB following three major volcanic eruptions. Runoff losses from the ice sheet were 280(±28) km^3 yr^-1 in 1961-90 and 391(±39) km^3 yr^-1 in 1998-2003. Significantly rising runoff since the 1990s has been partly offset by more precipitation. However, our best estimate of overall mass balance declined from -3(±53) km^3 yr^-1 in 1961-90 to -65(±61) km^3 yr^-1 in 1998-2003. Additional dynamical factors that cause an acceleration of ice flow near the margins, and possible enhanced iceberg calving, may have led to a more negative mass balance in the past few years than suggested here. The implication is a significant and accelerating recent contribution from the ice sheet, about 0.22 mm yr^-1 over the last six years, to global sea-level rise. Runoff and thinning of the ice-sheet margins increased substantially since the 1990s. However, massive snow accumulation over south-east Greenland during winter 2002/03, well shown in our analysis, led to unprecedented thickening in recent NASA aircraft LIDAR surveys. Do these recent changes indicate more extreme weather conditions including warming over the Ice Sheet, more storminess and higher accumulation events, due to global warming?