This genus is a problem. Nominally five species occur per Kew however in 2010 Masuyama and Watano showed with fairly robust DNA evidence that the plant we refer to as C. thalictroides is actually three cryptic species that do not interbreed. Thus, these three are listed here as well. Showing examples of these species is nearly pointless and should be considered rough guides. The genus is so polymorphic it my be possible for any species to look like any other under some conditions and only DNA or microscopic examination of the spores can tell them apart. It's not likely we have identified all species in this genus yet, it's increadably prolific.
The genus is found world over. Reports of the plant from Michigan, USA and Europe are almost certainly escapees from the aquarium trade, nominally the plant is of Asian origin although it is also found in Africa and South America; it's a primitive plant and was around long before the continents split.
Usually the pictures here are meaningful representations of the species. For this genus you can very nearly swap all the photos around and they are still correct. Don't read too much into identifications here, any species can look like all of these different forms in most cases.