So, I got this plant from a shop where there was a bit of a language barrier. But I was under the impression that it was some sort of epiphyllum and was expecting it to eventually flower with the typical big, beautiful flowers. Well, I am proud to say that I got it to flower... and the flowers are tiny. I love it anyway but now I am wondering what plant I actually have. Any ideas? The new branches are triangular (top right photo), but they turn into three distinct blades as the branch ages (middle right photo). Flowers are small, white-pink, and numerous (bottom right photo).

Cool plant!
Compare to:

There are more pictures in the Plant Data Base, but none showing the flowers. When I googled I found some pics of the flowering plants. This is just a guess. Hope some others chime in.

Yes, it is Rhipsalis paradoxa. Rhipsalis, I believe is jungle cactus. R. paradoxa is most attractive in a hanging basket as the stems can grow very long and form a drape. Good buy!