You can use “grandir” (to grow):

Les enfants sont toujours impatients de grandir, de pouvoir se coucher tard et conduire une voiture.

Of course it is more vague than the English “to grow up”, but the precision will have to be given by the context.

In addition to “grandir” given by Anne Aunyme, you could use “vieillir” (literally “to age”, “to grow old”)

See this definition (in french), or this one (in english).