Observations using COBE, SCUBA and ISO suggest that a significant fraction of the energy emitted by galaxies over the Universe's history was absorbed by dust and re-radiated in the far-infrared waveband. I will describe the evidence, discuss examples of the high- redshift galaxies detected, and investigate the consequences of the existing surveys for the history of star formation and the evolution of AGN. Direct observations of distant dusty galaxies will become much easier with the launch of SIRTF, and the development of next-generation ground-based submm-wave instrumentation, ultimately the commissioning of ALMA. The prospects for future observations in the submm and far-infrared will be discussed.