  We solve the existence problem for $F$-designs for arbitrary $r$-uniform
hypergraphs~$F$. This implies that given any $r$-uniform hypergraph~$F$, the
trivially necessary divisibility conditions are sufficient to guarantee a
decomposition of any sufficiently large complete $r$-uniform hypergraph into
edge-disjoint copies of~$F$, which answers a question asked e.g.~by Keevash.
The graph case $r=2$ was proved by Wilson in 1975 and forms one of the
cornerstones of design theory. The case when~$F$ is complete corresponds to the
existence of block designs, a problem going back to the 19th century, which was
recently settled by Keevash. In particular, our argument provides a new proof
of the existence of block designs, based on iterative absorption (which employs
purely probabilistic and combinatorial methods).
  Our main result concerns decompositions of hypergraphs whose clique
distribution fulfills certain regularity constraints. Our argument allows us to
employ a `regularity boosting' process which frequently enables us to satisfy
these constraints even if the clique distribution of the original hypergraph
does not satisfy them. This enables us to go significantly beyond the setting
of quasirandom hypergraphs considered by Keevash. In particular, we obtain a
resilience version and a decomposition result for hypergraphs of large minimum
degree.
