[grecy]

..and if your passport doesn't let you into country X, the airline won't let you board.

i.e. Russians need a visa to enter country X, you don't have one.

[guard-of-terra]

I don't recall airlines checking passenger visas. Either they don't or I'm not observant enough.

[grecy]

They check your passport, and they know if holders of passport from country X need a visa for country Y.

I've traveled quite a lot, and I'm always asked if I have authorization to enter the country I'm flying to.

[guard-of-terra]

I've travelled quite a bit (including Russia -> EU), and I've never been asked that.

Maybe they do a fast peek at your passport when letting you to board in the very end, but they can only check if some visa is physically there, since it takes them around 5 sec.