  We prove several results about p-divisible groups and Rapoport-Zink spaces.
Our main goal is to prove that Rapoport-Zink spaces at infinite level are
naturally perfectoid spaces, and to give a description of these spaces purely
in terms of p-adic Hodge theory. This allows us to formulate and prove duality
isomorphisms between basic Rapoport-Zink spaces at infinite level in general.
Moreover, we identify the image of the period morphism, reproving results of
Faltings. For this, we give a general classification of p-divisible groups over
the ring of integers of a complete algebraically closed field in the spirit of
Riemann's classification of complex abelian varieties. Another key ingredient
is a full faithfulness result for the Dieudonn\'e module functor for
p-divisible groups over semiperfect rings (meaning rings on which the Frobenius
map is surjective).
