Often theists assert that there is a forced choice on this subject, that if one cannot come up with a viable alternative theory for the origins of the universe, one must perforce accept that God created it. However, this notion has several problems.
The most obvious problem is that if the existence of the universe requires an explanation, then the existence of God (a literally infinitely more complicated entity) must also require an explanation.
If one says "God always existed", I have to ask why the mass/energy that makes up the universe could not have always existed, too. Why not save a step? After all, so far as I can see, mass/energy fulfills the properties we'd expect. Energy can be converted to matter, and vice versa - but we've never  seen mass/energy disappear, or spontaneously appear. Isn't that what we'd expect of something that has always existed?
On the other hand, some theists insist that God is 'by definition' simple, that It's 'one substance' with no 'parts' and in that sense not a 'more complex' explanation for the complexity of the universe. Of course, if they agree that the stupendous complexity of the universe can come from something perfectly simple... why do they try to argue against the sufficiency of things like evolution?
Also, even if one accepts that some kind of supernatural force must have created the universe, I see no compelling reason to assume that it must have been the Judeo/Christian/Islamic God. For example, why only one God; why not a pantheon? Certainly the platypus (an aquatic mammal with a duck-like bill that produces venom, has an electrical field sense, and lays eggs) looks like it was designed by committee.
And, of course, as the problem of evil suggests, there are strong reasons for believing that the J/X/I God would not have produced this universe. Many different types of deities have been designed to get around the 'problem of evil' that look nothing like the 'standard model' of 'God'. For example, some thought that this universe was actually made by Satan when God wasn't looking, or that God was not, in fact, omniscient. (For more information, see here.)
When it comes to the ultimate origin of the universe, I'm fine with saying "I dunno." I'm also content to say, "It looks like nobody else knows, either." Our brains don't like infinite regressions. (Though it's worth noting that they don't have any problem with infinite progressions, with the idea that time can continue infinitely into the future.) But our brains also don't like things popping in from nowhere. Aquinas et. al. couldn't think of any alternatives but those two, and decided they disliked infinite regressions more. But that doesn't mean they were right.
In the late 1700's, as scientists started getting a handle on electricity, they realized that lightning was electrical, and should respond in the same ways as the electricity they generated in their labs. Lightning rods were proposed, and the officials of various churches vociferously denounced them. After all, they knew that lightning was a direct expression of Divine fury, and it was hubristic to attempt to interfere with that. (You can see a hint of this sentiment remaining into the early 1800s in Herman Melville's story, "The Lightning-Rod Salesman".)
Of course, since God wouldn't strike a church with lightning, very often people would store explosives in the local church (the tallest building in town, with ungrounded metal on top). After the Church of San Nazaro in Brescia, Italy was struck by lightning in 1769, and 3,000 people were killed when 100 tons of gunpowder stored there exploded, those objections began to die out.
Now, even before the 1700's, was it reasonable to say that God (or Thor, or the Thunderbirds, or Zeus, or Seth, or what have you) caused lightning? No, the proper response to "What causes lighting?" was "Darn if I, or anyone else, knows."
William Thomson (Lord Kelvin) calculated that the Earth couldn't be more than a few million years old, because the Sun, which was radiating heat from gravitational collapse, couldn't emit energy as it was for longer than that. Of course, he didn't know about nuclear fusion...
Until we got a handle on gravity's behavior, and orbital dynamics, and so forth, meteors and comets were incomprehensible. Until Pasteur had his insights on germs, much of the workings of disease were incomprehensible. And all of these things were, until these insights were achieved, asserted to be 'obviously' the result of supernatural forces.
Until Watson and Crick (and Maurice and Franklin) worked out the structure of DNA, cellular reproduction and molecular biology was incomprehensible:
"What intelligible account can the mechanistic theory of life give of the... recovery from disease and injuries? Simply none at all, except that these phenomena are so complex and strange that as yet we cannot understand them. It is exactly the same with the closely related phenomena of reproduction. We cannot by any stretch of the imagination conceive a delicate and complex mechanism which is capable, like a living organism, of reproducing itself indefinitely often." - J. S. Haldane, 1932
Of course, now that we've had the critical insights about what DNA is and how it works, these phenomena are no longer so mysterious. (More on Haldane here.) The point is, it doesn't appear that anyone's had the right insights to make sense of the origin of the universe... yet. I don't think anyone has a complete, compelling theory about how the universe came about. I'd certainly like one, but I'm willing to accept that there are some things I may never know.
I like this passage from Bertrand Russell:
"[W]here you can get down to any knowledge of what atoms actually do, you will find they are much less subject to law than people thought, and that the laws at which you arrive are statistical averages of just the sort that would emerge from chance. There is, as we all know, a law that if you throw dice you will get double sixes only about once in thirty-six times, and we do not regard that as evidence that the fall of the dice is regulated by design; on the contrary, if the double sixes came every time we should think that there was design. The laws of nature are of that sort as regards a great many of them. They are statistical averages such as would emerge from the laws of chance; and that makes this whole business of natural law much less impressive than it formerly was."
Yes, the universe is quite complex, but we already know ways for very complex order to arise without design - if things like snowflakes don't impress you, go actually study evolution for a while and see what kind of amazing things are possible.
Evolution is a particularly good example of things that appear, to naive human intuition, to be 'designed' arising without need for a designer. Quick test: if you hear people describing evolution as saying that things arose 'just by chance' you know they haven't a clue what they are talking about. Chance does play an important role in evolution, but it is very far from the only component. If you don't understand this, go hip yourself now. You'll be glad you did. For specifics on this particular misconception, see here, here, and maybe even here.
(Oh, and BTW, before you accuse me of just accepting evolution on 'blind faith' or something, go look over here at a program I wrote myself. I have direct experience with evolution coming up with designs that I did not, and would not have, come up with.)
Some people talk about how certain critical physical constants must have the values they have for life to be possible at all. I'm not so convinced that all possible variations would be so devoid of any kind of life, but be that as it may, I'm also not convinced by this argument.
For example, it was once considered a physical constant that water freezes at a particular temperature (32 degrees Fahrenheit, 0 Celsius, 273.15 Kelvin) at normal Earth atmospheric pressure. Of course, we now know a lot more about how water is put together and how it interacts with itself and other molecules, and we can see why it freezes at that particular temperature, and why it couldn't be anything but that temperature, given what water's composed of.
I rather suspect that, with time, we may well find out why other 'constants' are the values they are, and may well discover that they couldn't logically be otherwise. In any case, Russell goes on to say:
"Quite apart from that, which represents the momentary state of science that may change tomorrow, the whole idea that natural laws imply a lawgiver is due to a confusion between natural and human laws. Human laws are behests commanding you to behave a certain way, in which you may choose to behave, or you may choose not to behave; but natural laws are a description of how things do in fact behave..."(Emphasis added.)
As I note here, if we find a rock that falls up, it means the "Law of Gravity" needs to be changed. The law is wrong, not the rock! "Natural laws" have changed in the past and will change in the future. They represent our best current understanding, that's all.
 Well, technically, there's Quantum fluctuation, but that's not really of much comfort to creationist types.