Tu or vous: who suggests it, and when
It is one of the most baffling features of the French language. The default rule, who proposes the switch, and the circles that are exceptions: everything you need never to hesitate again.

A business dinner, dessert arrives, and the person opposite you suddenly says, with a smile: “Shall we say tu to each other?” The moment is tiny, and yet it changes the relationship. French is one of the few European languages in which this boundary remains so alive, and where crossing it is a small social event.
The default rule
Vous is required with anyone you do not know, with older people, in shops, in public offices, and in most professional relationships. It is not coldness; it is a mark of respect, a way of not presuming an intimacy that does not yet exist.
Tu, on the other hand, is naturally used with children, family, friends, and often with colleagues of the same level in companies where that is the custom.
Who suggests it?
This is the heart of the matter. Traditionally, it is up to the older or more senior person, or to the woman, to propose switching to tu. The phrase is simple: “On peut se tutoyer, si vous voulez.” It leaves the other person free to reply warmly, or to stay politely with vous without anyone losing face.
If it has not been offered to you, you do not take the initiative of saying tu to a superior, an elder or a client. And if, carried away by the conversation, you let a tu slip by mistake, correct yourself with a smile, without making a scene.
No going back
Once adopted, tu is never withdrawn. Returning to vous after using tu would be felt as a break, almost an insult. That is why people think before suggesting it: switching to tu is a commitment.
Customs that vary
France is not uniform on this point. The worlds of start-ups, advertising, fashion and sport switch to tu very quickly, sometimes from the first minute. Some large companies have even made it an internal rule. Conversely, in some families of bourgeois or aristocratic tradition, vous is kept for a lifetime, between spouses or between parents and children. It is rare today, but it still exists, and it is not a lack of affection.
General de Gaulle addressed his wife as vous, and he was far from an exception in his generation. Using vous between spouses was long a form of respectful tenderness, not distance.
Madame, Mademoiselle
Finally, a word on a related question. Today, every adult woman is addressed as “Madame”, whatever her age or marital status. Since 2012, the French administration has even removed the “Mademoiselle” box from its forms. Used with an adult woman, “Mademoiselle” can seem condescending, even inappropriate. It is reserved for little girls and very young women.
In other languages
Italians have Lei, Spaniards usted, Germans Sie, with customs often even stricter than in France, particularly in the German-speaking world. The English now have only you, and distance is marked in other ways: by first name or surname, by tone, by a thousand nuances that foreigners struggle to grasp.
When in doubt, say vous: it is always easier to move closer than to step back.