I seem to remember someone using this as a farewell, but when I search for the expression, it doesn't seem to be as common as I thought. Is it idiomatic to say something like "До следующего раза"? If so, can one leave out "раза"?
In French, for example, one can say, "À la prochaine", meaning "Until next [time]".
I haven't encountered its use in this sense.
Usually it's said when some business is left undone and is expected to be completed or resumed next time around.
What can be said though is до скорого! (equivalent of the German bis bald) which is a short form of до скорого свидания which in turn is a variant of the conventional до свидания, or the straightforward до встречи!
Still in modern language до скорого is not particularly colloquial. Sounds a bit pretentious to my taste.
Expecting to meet a person on a known day or at a known time of the day people might say до вечера!, до завтра!, до пятницы!, до следующей недели!.
Typically you would hear the unspecific in terms of time пока!, давай!, на связи! (keep in touch), до связи! (talk to you later), счастливо!, удачи!, увидимся! (see ya) and their combinations.
Пока! does have a connotation of until, etymology was discussed in Russian at "Почему прощаются "пока"?" and at "Пока / Давай при прощании"
"До следующего" feels incomplete and wrong.
"До следующего раза" means not "until next time", but rather "until next occasion/event". As a farewell, it is OK to say to a person/a group of people who you meet only on some rare/specific occasions, according to some schedule, tradition, when the stars align, etc.
For example, you meet some people, say, once, twice a year for a party, a travel, something like that. When parting at the end of the event, because you don't expect to meet them until the next such event, you may say "ну, до следующего раза" (e. g. "until the next party/whatever").
