The flaws in the Argument from Design are several. First, using the existence of complexity as a proof for God amounts to a self-contradiction; second, a common form of this argument (made famous by William Paley) misunderstands how humans identify intelligent design; and third, a common version of this argument that is based on probability misrepresents the role of randomness in evolution.1 Let's look at each of these in turn.
Setting aside any appeal to Darwinism for the moment, what could it possibly mean to say that complexity in living things implies the existence of an intelligent designer like God? One can only assume that God, whatever that term might refer to, must have at least as much complexity as anything He is supposed to have designed. Given the theist’s assumption that complexity requires a designer, God’s own complexity implies that He also had a designer. Either the theist is arguing for an infinite regress of God-designers and designers of God-designers, etc., or he is contradicting his own assumption that complexity requires design. By using God as an “explanation” the theist is doing nothing more than explaining complexity (in living things) with complexity (God’s). But this amounts to assuming what one is trying to explain, which is no explanation at all. It just moves the mystery back a step.
This is the same logical flaw in using God to explain existence itself. The theist often asks, “If you don’t believe in God, then how do you explain the existence of the universe?” This question assumes that existence must be caused, and since the universe clearly exists, it too must be caused. The theist then concludes that God must be that cause. Now, presumably the theist supposes that God, like the universe, also exists, in which case the theist is right back to violating his own assumptions: If God exists, and existence must be caused, then by the theist’s own assumption, God must be caused. By using God as an "explanation" the theist is doing nothing more than explaining existence (the universe's) with existence (God's). And just as before, this amounts to assuming what one is trying to explain.
Typically the theist’s reply to these criticisms is that God is the one exception: All complexity except God’s complexity must be explained, and all existence except God’s existence must be explained. But this is blatant special pleading. The theist is simply exempting himself from his own rules: "Your explanation must meet these conditions; however, my explanation (God) does not." Of course, anyone can play this game. Once could just as easily (and with considerably more parsimony) say all things except the universe as a whole require an explanation.