In Russia (at least when I visited in 1995-96) there are "routed taxis" (marskrutnaya taksi) and "taxis". The routed taxis are vans and just take you to the nearest metro station. They fill the gaps in the streetcar/trolleybus/autobus network in the outer parts of the city.
The "taxis" are private cars, driven by people who need money or are just bored, so they drive around all night giving people lifts for maybe $3 a trip. People who don't want to be seen taking public transportation take "taxis" instead, and everybody uses them in the off-hours. You just hold out your hand (straight out, not thumb up) at the passing cars until one stops, and then negotiate a fare. I had friends who drove them occasionally. I wondered if they were safe but I never heard of anybody getting robbed in one.