If there is one argument for the existence of God which stops and makes me pause it is the fine tuning argument. While I am not impressed with the logic of the conclusion, there are a number of things about this universe which seem to require an explanation.
That the universe has produced intelligent creatures which are able to figure it out (at least in a limited sense) seems highly implausible given our current present understanding of physics (those parts of physics which have currently been verified, as opposed to speculative M-theory, etc).
Some might say that the fact that the universe has produced intelligent creatures is of no more significance than a universe filled entirely with black holes - although this is technically true in a value-neutral sense, there is still something non-mundane about creatures which have the ability to create value. One might also say that the relative complexity of the human brain, which produces meaningful consciousness, is that which is to be valued. Regardless, I will assume that humans do have value.
That human brains have a level of complexity which can produce meaningful consciousness (higher level thought versus chimpanzees, etc) is a fairly uncontentious observation. The explanation of how humans arrived at this state however is contentious. Many would have us believe that only God can be responsible, while most scientists and atheists alike will say that evolution provides a satisfactory explanation.
Although humans have evolved their brain under evolution, it is not true that in all possible universes that evolution can even start. Under our current understanding of how the universe works, the universe needs the correct conditions to allow life to start at all. For example, some universes may only last millions of years instead of billions of years - which isn't enough time to allow life to form (life requires heavy elements which can only be formed in supernovae) and evolution to take hold all else being equal. Some universes may only consist of black holes (gravity too high) - such a universe would be unsuited for starting life, hence evolution cannot even start to produce complex beings.
Although there are many unitless constants which come up in different areas of fundamental physics, Marin Rees has come up with six constants which he deemed to be fundamental to the structure of the universe. These are generally the values/constants quoted by those who defend the fine tuning argument for theism and I will use them here.
Perhaps the most significant of these constants is the cosmological constant which is said to be fine-tuned to within 120 orders of magnitude. (Steven Weinberg 1989 "The Cosmological Constant Problem") That is to say that if the value were increased by 1 part in 10^120 the universe would have expanded too rapidly.
1. That these six numbers are "dials" a designer/universe generator can play with (or degrees of freedom for possible universes).
2. That the laws of physics as we know them must apply to all possible universes.
3. That we know the possible probable range the six numbers can take.
4. That we understand physics enough to come to a reasonable conclusion (i.e. god-of-the-gaps)
5. That there is only one possible universe.
1. Currently there is no grand unification theory or any theory which is able to explain the laws of physics as we currently know them. Given this we cannot say whether these constants are actually constants or if they are based on more fundamental relationships, either mathematical or on some other "constant".
2. We only know the laws of physics given that which we see in this current universe. What reason do we have to suppose that the laws of physics are the same in all possible universes? Are the laws of the universe like Pi and a logical necessity or are they merely contingent.
3. Unless we have a defined probability of the range of values our fine tuned constants can take - our set of fine tuned constants are actually isomorphic to a coarsely tuned universe.
4. Our knowledge of physics isn't sufficient to come to grand conclusions about the nature of some yet-unknown unifying theory. In this sense, the fine tuning argument is merely a god-of-the gaps style of argument where it feeds on our current lack of knowledge.
5. Given that we only have one universe, we are unable to conclude anything about general possible universes based on purely empirical methods. That is to say we cannot say the possible ranges of these constants given only one universe without any other reason why. We cannot also rule out multiple universes outside of our detection abilities or current detection abilities.
Isn't it interesting how physics concludes that life is possible given the constants of nature as they are? God could have created the universe in such a way to make life impossible under pure naturalism (a 6000 year old earth, etc), yet apparently we are to believe that he did so in a way which still makes sense under naturalism. Wouldn't it be miraculous if the universe couldn't have possibly have created life - yet still did. THAT would require a serious explanation, however the fine-tuning argument as-is isn't a threat to the naturalists ontology.