Answer:
Betta fish are about the only kind of fish that can survive with no filter. They prefer still water or a very gentle current, so as long as you are keeping their water warm and clean (at least 72* F) they will thrive in a one gallon, unfiltered bowl. Change the water at least weekly to keep a one-gallon betta tank clean. Change more often in anything smaller (and it's not recommended to keep them in anything smaller).
Many people think goldfish can survive without a filter, but this is false. Goldfish are one of the dirtiest fish out there and ammonia will quickly build up in a small, unfiltered bowl, severely shortening your fish's lifespan. A properly cared for goldfish can live for fifteen to twenty years and reach or exceed two feet long.
Fat bodied goldfish do prefer gentle currents due to the way their body shaped. Their round body and tiny fins makes struggling against a strong current difficult. But that does not mean they do not need one.