Astronomical light echoes, the time-dependent light scattered by dust in the vicinity of varying objects, have been recognized for over a century. Initially, their utility was thought to be confined to mapping out the three-dimensional distribution of interstellar dust. Recently, the discovery of spectroscopically-useful light echoes around centuries-old supernovae in the Milky Way and the Large Magellanic Cloud has opened up new scientific opportunities to exploit light echoes. In this review, we describe the history of light echoes in the local Universe and cover the many new developments in both the observation of light echoes and the interpretation of the light scattered from them. Among other benefits, we highlight our new ability to spectroscopically classify outbursting objects, to view them from multiple perspectives, to obtain a spectroscopic time series of the outburst, and to establish accurate distances to the source event. We also describe the broader range of variable objects whose properties may be better understood from light echo observations. Finally, we discuss the prospects of new light echo techniques not yet realized in practice. ; Comment: 18 pages, 7 figures, 1 table. Accepted for publication in PASA