Dawson Casting: Because the series takes place over nearly eighty years, this is both played straight and later inverted with many of the same characters. Just to name a few major examples:
Derek Jacobi, aged 38, played Claudius from the age of 19 to his death at 63.
Siān Phillips, aged 43, played Livia from the age of 34 to her death at 86.
George Baker, aged 45, played Tiberius from the age of 18 to his death at 77.
Margaret Tyzack, also aged 45, played Antonia from the age of 27 to her death at 73.
Some examples where this is purely played straight include John Hurt (aged 36) as Caligula from the age of 17 to his death at 28, Christopher Biggins (aged 28) as 17-year-old Nero, and Graham Seed (aged 26) as 13-year-old Britannicus.