In French, there are different ways to express the day after/the day before.
Like in English, demain (tomorrow) and hier (yesterday) are used to talk about moments considered from the point of view of the present.
Using "demain/hier", "le lendemain/la veille", "le jour suivant/le jour précédent" in French
Sometimes you talk about moments that are seen from a past or future point of view, i.e. when the speaker places himself within events that already took place or events that are further in the future.
In these cases, we use the following expressions to talk about the day after and the day before:
Le lendemain / La veille
Le lendemain generally means the day after or the next day.
-> You cannot say le jour après in French.
La veille generally means the day before, but it can also be used more specifically in the sense of the eve, for example at Christmas:
To emphasise that you're talking about the evening before or the morning before, you can also use the expressions la veille au soir and (rarer) la veille au matin, but it doesn't work with l'après-midi:
Le lendemain/La veille de + noun = the day after/before [something]
The main difficulty here is that in French you cannot use a conjugated verb after la veille or le lendemain, unlike in English: the day after he leaves / the day before you were born.
Instead you will use de + noun, as such:
Le jour d'après / Le jour d'avant
These can only be used on their own, and will mean the same as le lendemain and la veille, although they're a bit less elegant, more used in speech.
Note that this expression can also be with days of the week:
Le jour suivant / Le jour précédent
As for le jour suivant (on the following day) and le jour précédent (on the previous day), they are used in a past or future context just like le lendemain and la veille, but always on their own.
Note that just like le jour d'après and le jour d'avant, this expression can also be with days of the week:
Le jour suivant / Le jour précédant + noun = the day following / preceding [something]
To say the day following his arrest or the day preceding/leading to their first date, you will use le jour suivant or le jour précédant + noun.
See also Prochain/Dernier = Next/Last + [durations] in French
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Examples and resources
The next day, I was enrolling at university.