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We solve the existence problem for $F$-designs for arbitrary $r$-uniformhypergraphs~$F$. This implies that given any $r$-uniform hypergraph~$F$, thetrivially necessary divisibility conditions are sufficient to guarantee adecomposition of any sufficiently large complete $r$-uniform hypergraph intoedge-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 thecornerstones of design theory. The case when~$F$ is complete corresponds to theexistence of block designs, a problem going back to the 19th century, which wasrecently settled by Keevash. In particular, our argument provides a new proofof the existence of block designs, based on iterative absorption (which employspurely probabilistic and combinatorial methods).  Our main result concerns decompositions of hypergraphs whose cliquedistribution fulfills certain regularity constraints. Our argument allows us toemploy a `regularity boosting' process which frequently enables us to satisfythese constraints even if the clique distribution of the original hypergraphdoes not satisfy them. This enables us to go significantly beyond the settingof quasirandom hypergraphs considered by Keevash. In particular, we obtain aresilience version and a decomposition result for hypergraphs of large minimumdegree.

The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$