Well, very likely you do dream, but aren't remembering them. Now some people sleep so poorly that REM sleep is constantly being interrupted, but this is really not common. Dreaming seems to be a very necessary part of our daily lives actually. Its healthy, normal, and probably necessary for our mental health. How much more healthy if we paid attention to our dreams and understand them! But we have to notice them and remember them first.
Sometimes that means overcoming some biases and resistances. For example: dreams are "crazy" and a waste of time; they are scary; they can make me feel things I don't want to feel- and so on. Its true dreams don't follow the rules of the waking world we think of as "sane" and "normal". Dreams will put very odd things together sometime! And this can be disturbing! Its important to just accept dreams as they are and not judge them or yourself for dreaming such unique creations.
Dreams work mainly with images. The expression "a picture is worth a thousand words" comes into play here. One image, lets say a red hat, in the context of a dream can mean A WHOLE LOT if you start to consider it in the context of the dream. Who is wearing it? is it oversized? is it shabby or new? is it like the hat Aunt Mary wore? and a whole cascade of meaning can come from all this (see Associations). But in general, dreams deal in symbolic images and what is literally happening in the dream (referred to as the "manifest content") (that crazy stuff!) can make a whole lot of sense and have profound meaning when its work through.
We seem to go through different levels of dreams through the night: the late night dreams are sometimes more "cryptic" to us and working at a deeper level of consciousness while dreams close to morning can be closer to daily life issues and more easily accessible. Dreams can also come in series, or continue through the night like an ongoing play. As we pay attention to dreams they can change over time- in a sense they respond to our increasing understanding through dream work.
A useful habit is to keep a pad of paper next to the bed and without getting up, just write the dream as you lay there. Often as you start to write a dream you may begin to remember more details and even whole sections of the dream that you didn't recall right upon waking. Sometimes its best to just start writing what you remember and go from there. Of course it helps to indicate "this section came at the start" if it is written out of sequence.
I've been told it can help to go to bed with the intension to remember your dreams in the morning.