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              AN INTRODUCTION TO SUPERPROCESSES

                           Robert Adler

               Technion, Israel and UNC, Chapel Hill

Once upon a time, there was a random walk, which even undergraduates
understood. As time went on, it became normalised
in space and time, and, eventually, it converged to Brownian
motion. By then, only graduate students could
really understand what it was, but they, and many others, soon
realised that Brownian motion was a {\it good thing}.

For a start, Brownian motion was related to the heat equation,
and this was good because it helped probabilists understand Brownian
motion and because it helped analysts understand probabilists.

More recently, there was a branching random walk, which, using their
experience,  probabilists immediately ran in speeded up time, and
normalised in both space and time. The result, they (eventually) agreed
to
call {\it super Brownian motion}. Immediately, the world of
probabilists,
led by the Markov theorists, realised that super Brownian motion was
also a {\it good thing}.

This, primarily expository, talk is designed to show why such is the
case.