Betting games with variable stakes, called martingales, can be
used to define infinite sequences which are random or
pseudorandom with respect to a given computational resource
bound. Recently Buhrman and Longpre have shown that a
martingale can be interpreted as an algorithm for compressing
the sequences on which the betting strategy is successful, and
that a sequence is incompressible with respect to a given
resource bound if and only if the sequence is random with
respect to the same bound. Our main goals in this talk are to
provide an elementary introduction to martingales and resource-
bounded probability, to discuss the incompressibility
characterization of randomness its possible relationship with
certain "practical" data compression algorithms (arithmetic
coding), and to discuss some new results describing conditions
under which a sequence can be maximally compressed.