  Random substitutions are a natural generalisation of their classical
`deterministic' counterpart, whereby at every step of iterating the
substitution, instead of replacing a letter with a predetermined word, every
letter is independently replaced by a word from a finite set of possible words
according to a probability distribution. We discuss the subshifts associated
with such substitutions and explore the dynamical and ergodic properties of
these systems in order to establish the groundwork for their systematic study.
Among other results, we show under reasonable conditions that such systems are
topologically transitive, have either empty or dense sets of periodic points,
have dense sets of linearly repetitive elements, are rarely strictly ergodic,
and have positive topological entropy.
