Did time exist before the Big Bang and the creation of the universe?
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.
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.
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.
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.)