We survey some upper and lower bounds established recently on
the sizes of randomized branching programs computing explicit
boolean functions. In particular, we display boolean
functions on which randomized read-once ordered branching
programs are exponentially more powerful than deterministic
or nondeterministic read-$k$-times branching programs for any
$k=o(n/\!\log n)$. We investigate further computational
power of randomized read-once order branching programs
(OBDDs) and their basic manipulation properties for
verification of boolean functions and for testing graphs of
arithmetic functions.