Part of my fascination with these humble flatworms comes from their ability to regenerate - as a youth, I remember coming across drawings of two-headed flatworms, multiheaded flatworms, and full-sized flatworms with a 'miniature' flatworm emerging from their sides. Not all flatworms display this amazing ability to regenerate, of course - depending on which version of flatworm taxonomy you accept, regeneration is thought to be an ancestral characteristic that has been reduced or lost multiple times in the evolution of this group. Some flatworms can't regenerate at all, others can regenerate most of their body, as long as the brain is intact, but for most flatworms regeneration of amputated parts tends to be limited to an area from their brain to their mouth (located around the middle of their body). They do a bit better at regenerating cuts, which leads to two-headed flatworms if you make a cut along their midline.
The flatworms that do regenerate have a cells scattered through their body called neoblasts that are thought to be totipotent - the flatworm equivalent of stem cells. When a flatworm is damaged, these cells migrate to the wound site and begin dividing and differentiating to make new tissues. What is interesting about this process is why not all flatworms regenerate, and why some are not as good at it as others even though they have neoblast cells... if you think about it, these cells are a two-edged sword. On the one hand, they allow the animal to quickly repair damage and replace lost body parts (and considering how soft and delicate these animal are, I'm sure damage is quite common), on the other hand, these cells have to be kept under tight regulation or they run the risk of replicating out of control - essentially becoming cancer cells. Even a regeneration that is 'mostly' correct can be fatal - two headed flatworms appear to die after a month or so. I suspect that for some lineages of flatworms, the risks of uncontrolled regeneration or cancer have outweighed the benefits of fast healing and replacing amputated parts, and over time the ability to regenerate has either been lost outright or various mechanisms to regulate and restrict these cells have evolved.
Of course there are other factors that may help maintain the ability to regenerate. A number of flatworm species can reproduce by splitting in two, and while there are differences between fission and regeneration, neoblast cells are involved in both processes. Interestingly, in some of these asexual species, if they do become sexual and develop ovaries, they lose much of their ability to regenerate. There are also similarities between early growth and regeneration, suggesting that regeneration could be a retention in the adults of growth mechanisms in the juvenile. It would be interesting to look at some of the poor- and non-regenerating species and see if their juveniles show greater regenerative powers than the adults.
As for my own flatworms, I'm a bit undecided as to whether I will be cutting them up to create two-headed monsters of the water-tank... for one, cutting a tiny flatworm is probably a lot harder than it sounds if all you have is an old microscope, a hand-lamp, and a kitchen knife (ah for the days when I had access to a full laboratory!). For another, where people have bothered to study the fate of these multi-headed worms, their lifespans have been much shorter than normal, and for now I just want to see if I can keep them alive in a small tank. These aren't the 'classic' lab flatworms with the arrow-shaped heads (Dugesia tigrina), so I don't have as much to go on about culturing these species.