Identification: Help me identify my plant

We have this wonderful plant and have no idea what it is. It sprouts new leaves from existing leaves.
Can you please help us identify it?

 

An Epiphyllum cactus. The 'leaves' are actually flat stems: like most cacti, the real leaves are so small (and drop off so soon after being produced) that you don't notice them.

 

Thanks so much! Do you know how I can make cuttings off of it?

 

Quite easy. Just cut a shoot, and allow it to dry off for about a week so the cut base gets callused (the callus protects the base from getting infected with decay fungi), then put it in a leaf-mould based compost mix. Keep it on the dryish side at first, again to reduce the risk of decay.

 

Such spectacular blooms they have, too. This plant is on my "I WISH" list!