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.