C0-sleeping is when parents sleep in the same bed with their babies. It is also sometimes called share sleeping or family bed.
My husband and I have co-slept with each of our 5 children so far, and I would have it no other way. Our youngest is 2 months old and sleeps with me every day and night (including a nap each day, which I love). Particularly when the baby is very small, I find it very comforting to have my baby next to me where I can hear and feel his breathing and movements and know that all is well.
I've also breastfed all of my children, and night feedings are so much easier when I can just nurse the baby in bed with me. The baby never has to wake enough to actually cry because I am tuned in to his hungry cues even when we're both sleeping, and can simply put him to the breast and we both fall back asleep easily. I never have to get out of bed to go to the baby or prepare a bottle, and we both sleep better because of it. My baby nurses longer at night time feedings, but I think it's because he nurses less often during the night than during the day (about twice each night), and it doesn't bother me at all because we are usually just sleeping through the feedings anyway.
My babies have never slept well in cribs. They always sleep more soundly and longer when they sleep with me. On the few occasions I tried putting my baby in a crib with my first baby, I would wake up worrying about whether or not the baby was breathing, and would have to get up just to check, and it interrupted my own sleep quite a bit.
I've always been a deep sleeper, and as a teenager it took a huge amount of effort just to wake me. However, when I sleep with my baby I am always aware of where the baby is and instinctively know how to lie with him. I usually sleep on my side with the baby in front of me, and I feel like I'm being a protective shell as I can curl my body around him.
The following article in Mothering Magazine explores the cultural impact of co-sleeping:
The Complexity of Parent-Child Cosleeping: Researching Cultural Beliefs
An interesting discussion of individual viewpoints and the safety of co-sleeping:
Breastfeeding Discussion from Milkalicious on My Best Birth