Professor Peter Kreeft, a philosopher at Boston College, tackles this question in a brief five minute lecture. Since his talk is so brief it's intended to serve only as an introduction to a problem which has vexed philosophers and theologians for millenia. Even so, Kreeft's treatment gives us a good idea of the outlines of the difficulty and the direction in which possible solutions might lie.
One note: Kreeft refers to the matter as the problem of evil, but I think it's better to think of it as the problem of suffering. The word "evil" is a moral term which, I would argue, should have no place in the vocabulary of the antitheist who is employing the existence of suffering as a defeater for belief in God. The antitheist could easily evade this objection, however, by reframing the problem as a matter of suffering which carries none of the moral freight that the word "evil" does. Changing the word does nothing to diminish the force of the objection to theistic belief but does eliminate a potential sidetrack.
Ar any rate, here's Kreeft's presentation:
Thanks to HotAir.com for the tip.