Did time exist before the Big Bang and the creation of the universe? Does time stretch all the way back for infinity or was there a point when time appears to start in the universe?
I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe.
Does time have a starting point of note?
 A: In short, we don't know. There are a few indications that time started at the big bang, or at least it had some form of discontinuity. This might be wrong though.


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*According to General Relativity, there is no such thing as an absolute time. Time is always relative to an observer, without the universe there would be no corresponding concept of time. All observers within the universe would have their clocks "slowed down" the nearer they are to the big bang (nearer in time). At the big bang point, their clock would stop. This said, we know that GR doesn't apply as-is all the way to the Big Bang.

*Some cosmological theories like CCC predict a series of aeons and some form of cyclic universe. These predict a discontinuity (CCC predicts a conformal scale change) of time at the big bang, and at the end of the universe.

As a side note: people tend to have a special fascination with time. For all we know though, time is only relatively special. From a cosmological point of view the discussion is whether space-time existed. We are pretty sure that it was very very small at some point.
A: There is a simple answer: time is just a way of labelling different configurations of matter and/or space with a number. Therefore, there is no time without matter or space, or without change in matter/space. 
In other words, for there to be a concept of time, something has to be able to exist in different states. Imagine there was nothing but one solid sphere sitting in space, never changing (and no observer). There is no time in such a situation. The big bang singularity would be even more drastically of this sort.
(Meta-physically, some form of many-worlds theory could eliminate any autonomous meaning of time altogether, by assuming that all the different configurations of matter/space are 'there' somehow, and time just means going through them in a certain order.)
A: The only well tested theory of gravity we have right now is general relativity (GR). In models based on GR, time and space only exist for $t>0$. In relativity, we use the term "event" to mean a certain position in space at a certain time. The big bang is not an event, because there is no time $t=0$. If you want to find a cause for some event happening at a given time $t>0$, there is always some earlier $t'$, with $0<t'<t$, that can supply that cause. Since the big bang isn't an event, it doesn't have a cause. 
We also have fundamental reasons to believe that GR lacks self-consistency under the very dense and hot conditions at $t \lesssim 10^{-43}$ s (known as the Planck time), because of quantum-mechanical effects. If we had a theory of quantum gravity that worked under those conditions, then it might turn out that the singularity at $t=0$ was not real, and events at $t>0$ could be explained in terms of causes at $t<0$. This is what seems to happen, for example, in loop quantum cosmology. However, nobody has a theory of quantum gravity that  works and has been tested against experiment, so we don't really know.
"I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe." I don't think there is any professionally researched scientific theory that says this. Theories that have time before the Big Bang generally do not have a beginning to time at all.
A: Really, the question does boil down to metaphysics if taken out of the mathematical construct. It's a matter of simple metaphysical logical consistency. Time cannot have a beginning because time cannot, by definition, be a function of NO time. But it cannot be eternal (without beginning or end) either because any value of time given to an object on the timeline must logically be infinity. Also, static time is a contradiction in terms. 
I am ceding that this is a metaphysical question like others have done. Down vote is puzzling. I have edited. 
A: What is time? There are a number of possible answers, but none definitive https://en.m.wikipedia.org/wiki/Arrow_of_time
The universe gives us observations, which are facts. We return it theories, which are interpretations with predictive power over patterns in the facts. Causality, and time, may only be fundamental to how we organise our experiences, and not fundamental to the world. That is, interpretive not factual. 
The significance of that, is it may only be only our consciousnesses which travel in one direction through time,  like balls roll down hill while the hill is there all the time (manyworlds probability tree, general relativities block universe, or something inbetween). 
There are various categories os cosmological speculation about before the Big Bang, as opposed to time itself https://en.m.wikipedia.org/wiki/Big_Bang#Speculations
A: 
Does time stretch all the way back for infinity or was there a point when time appears to start in the universe?

Time is infinite, both ways. If it started from some point, then you can always assume that there was still lot of time before that "some point".

I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe.

That is our (human's) perception of time. But you can always imagine a mysterious clock which was still ticking while and before big-bang.

Does time have a starting point of note?

No. Time is eternal, hence no starting or ending points.

One may ask, if the time is eternal, then that means we have infinite past. So how come we reach to the present moment?  
That seems to be philosophical question, but it can be proven logically as well. For now, we may think that there were infinite cycles. All cycles were identical. Due to infinite cycles, time becomes infinite. Due to all of them being identical, at some point we can exist in anyone of them; because length of cycle is finite.
In other words, for example if big bang was start then big crunch is end of the cycle. That includes all the events happening in between. These 1 finite length cycle would keep repeating perfectly.
