  We solve the existence problem for $F$-designs for arbitrary $r$-uniform
hypergraphs $F$. In particular, this shows 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 $G=K_n^{(r)}$ into edge-disjoint copies of $F$, which answers a
question asked e.g. by Keevash. The graph case $r=2$ forms one of the
cornerstones of design theory and was proved by Wilson in 1975. The case when
$F$ is complete corresponds to the existence of block designs, a problem going
back to the 19th century, which was first settled by Keevash. More generally,
our results extend to $F$-designs of quasi-random hypergraphs $G$ and of
hypergraphs $G$ of suitably large minimum degree. Our approach builds on
results and methods we recently introduced in our new proof of the existence
conjecture for block designs.
