It was one year ago that my sister gave me the lovely gift of a male and female stick bug. No I am not being sarcastic, I had really wanted some ever since I saw the ones she had some years before. But no pet stores carried them, and my occasional Craigslist inquiries for them had gone unanswered.
Here is my original pair from last October (the male is darker, at the top) - click any photo for a slightly larger image:

The male is still alive, one year later.
The female started dropping eggs (to the bottom of the cage) 5 months after I got her, an egg or so every couple of days. She dropped 25 eggs total, and then died (of exhaustion?). I kept the eggs moist (by lining the bottom of the cage with paper towels, which I poured water on). The eggs started hatching about 2 months after they were dropped. Some 18 or 19 of those eggs hatched. Here is a newly hatched stick:

Some of the babies are now the size of the stick bugs I received last October. In fact, I have a hard time now telling which male is the father, and which ones are the babies.
I will keep a few stick bugs to continue the cycle, but I intend to adopt out the remaining ones. My sister happily took 2 females and 1 male. As of my count today, while cleaning the cage, I have 12 females and 5 males, which is too many to keep, once the females start laying eggs.
As I understand it, the females can reproduce without any males. In this case, all offspring will be daughters, and identical to their moms. But with a male, the offspring will be mixed males and females (as in my case), and with some genetic variation. For example, some of my females have a bit of green colouration on their legs, and others are brown throughout.
They are very easy to care for, their diet (and they don't eat very much) consisting of either oak leaves or blackberry leaves. Their maintenance consists of bringing in fresh blackberry leaves. I keep mine in small jars of water, so they only need to be changed every week or two, and I have no shortage of blackberry vines invading my yard from two sides. I always manage to poke myself as I try to change over the blackberries and move the stick bugs over from the old branch to new. But not seriously, yet.
I keep my sticks in a reptile cage, which has a mesh top, and doors which swing open and close with a small latch. It is a 18" cube. A smaller cage would do, since the sticks really don't move much, and don't eat much either. But this one provides great viewing of them.

When there are no eggs, the cage just needs to be cleaned occasionally, since the sticks drop little dry poops all over. I line it with paper towels, so I can remove them easily, and replace them. When there are eggs, they need to be kept moist. I use a few layers of paper towels, and pour a bit of water on them every day or two.
Apparently if one doesn't want the eggs to hatch, they should be frozen first (to sterilize them), before being discarded. I didn't need to do that, at least on this first cycle.
The stick bugs don't bite. They are gentle. They tickle. I think they make a great "pet" for children. Except that kids eventually get bored of them. Mine did. Now they're just my pets again.
If anyone from the Vancouver, BC area is interested in adopting some stick bugs, let me know.
30Oct09 Update : Happily, I have found some good homes for my sticks, and am down to a more reasonable number now:
1 male and 2 females to my sister
1 female to Susanna, mother of two boys in our school
1 male and 1 female to Beth, mother of a girl (good friend of my daughter) and boy in our school
1 male and 1 female to a school teacher in my sister's school, who is apparently "into" lots of different types of creatures, which she keeps in the classroom - cool!