saturday night stay is the airline requirement that you must stay over a Saturday night during a Trip in order to obtain the lowest fare.
Usually, there is also an Advance purchase requirement as well as other rules that apply, such as travel being limited to certain days of the week, etc.
Other definition:
Saturday Night Stay is an airline requirement that passengers must stay over a Saturday night during a trip in order to obtain the lowest fare. It looked like this requirement was disappearing, but many airlines have brought it back.
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