Our best theory of gravity is classical General Relativity. "Classical" is physics-speak for not taking into account quantum mechanics. So we know that classical GR has to break down during the Planck era, if not later.
Classical Big Bang cosmology predicts that there is an initial singularity at the first moment of time. In fact, there are some theorems to that effect. These are the Penrose and Hawking singularity theorems, which will be the subject of this post.
In GR, attractive gravity is caused by the energy or pressure of matter. Tension (which is negative pressure) produces antigravity (repulsion rather than attraction).
Very crudely speaking, the singularity theorems say that if you assume that matter obeys some energy condition restricting the amount of energy and/or pressure, then you can deduce that under certain conditions there has to be a place where your spacetime has an edge and cannot be extended any further. This we call a singularity. Typically, some component of the curvature becomes infinite at the singularity.
There are several different singularity theorems, pioneered by Hawking and Penrose. One of them says that singularity theorem says that all expanding cosmologies like our own have to begin with a singularity. Roughly speaking, it says that if there is only gravity and no antigravity, then tracing the universe backwards in time there is no way to stop it from crunching down to zero size. Hence there must exist an initial singularity (at least somewhere, perhaps everywhere).
However, this Hawking-Penrose theorem uses something called strong energy condition, which says that the repulsive antigravity from tension is not allowed to be greater than the gravity from energy. It turns out that the strong energy condition can be violated by lots of different types of otherwise reasonable physics theories. In particular, it was violated during inflation, and it is violated by the cosmological constant today. So no one really takes this theorem very seriously anymore.
There is another singularity theorem (proven originally by Penrose) which is better, because it only uses the null energy condition, which says that nearby lightrays are always focused by gravity. This turns out to be a much weaker condition, which is satisfied by most respectable classical matter theories (although it is violated quantum mechanically). However the Penrose singularity theorem only says that there has to be a singularity if space at one time is infinite.
If space at one time is finite in size (for example, if it is shaped like a 3-sphere) then there might be a "bounce" where the universe contracts to a small size and then starts expanding again. The de Sitter cosmology is an example of this, although there are also examples of finite cosmologies that begin with singularities. We don't really know whether space is finite or infinite, since inflation stretched it out so much that even if it were a giant sphere, the radius is so large that it seems flat today.
A few years ago I wrote an article in which I argued that the conclusions of the Penrose singularity theorem should continue to hold in quantum gravitational situations. Even though the null energy condition can be violated by quantum fields, it turns out that you can get the same conclusions if you instead assume something called the "Generalized Second Law" (GSL), which says that the Second Law of thermodynamics applies to black holes and similar types of horizons.
(I described the application of this result to time travel in a recent Scientific American blog post. Technically, you have to use the time-reverse of the GSL, which I mentioned in the comments here, but if the GSL is true, its time-reverse should also be. This may seem weird because normally we think of the Second Law as something which only works in one time direction, but I promise you that one can make sense of it.)
The advantage of using the GSL is that it makes it more plausible that the conclusions of the Penrose singularity theorem apply even in fully quantum-gravitational situations, e.g. during the Planck era. In my article, I showed that the results apply "semiclassically", meaning when the quantum corrections to spacetime are small but still taken into account. I then argued (and not everyone would find this part of my article persuasive) that under certain assumptions one might expect the result to hold even in full quantum gravity, when these quantum fluctuations are large. But remember, all statements about quantum gravity are speculative.
I am a little reluctant to even bring up my own work, since personally I think it is more persuasive that clearly established (but incomplete) physics predicts a beginning, than that speculative new physics says this. I think of it more as laying the groundwork for a possible future understanding, then a totally conclusive result. Still, I think that the Penrose theorem is connected to enough other deep principles of physics that something like it will probably be true and important in the final theory of physics. Other physicists think that singularities are so disturbing that any "complete" theory of physics should eliminate them.
Funny story. One time I was arguing with an atheist grad student about God and the question of the universe's beginning came up. I mentioned my own work (and I am quite sure that I put in some caveats about the potential limitations, since I always do this). A few weeks later I found him posting on some atheist website cocky statements along the lines of "Theists believe that the universe had a beginning because of the GSL, but this is silly for the following reasons...". And this at a time when practically no one else had even heard of my work! So just in case it isn't clear: many smart people believed in God before I came along, and the case for Theism is hardly dependent on my tiny contributions to physics!
In conclusion, to the extent that the singularity theorems are relevant, they tend to point to a Beginning, although it might be possible to evade this conclusion either by (a) having space be finite, or else (b) through quantum gravity effects, if my speculative arguments for a quantum singularity theorem are wrong.