Summary: A fundamental theorem of Wilson states that, for every graph \(F\), every sufficiently large \(F\)-divisible clique has an \(F\)-decomposition. Here a graph \(G\) is \(F\)-divisible if \(e(F)\) divides \(e(G)\) and the greatest common divisor of the degrees of \(F\) divides the greatest common divisor of the degrees of \(G\), and \(G\) has an \(F\)-decomposition if the edges of \(G\) can be covered by edge-disjoint copies of \(F\). We extend this result to graphs \(G\) which are allowed to be far from complete. In particular, together with a result of F. Dross [SIAM J. Discrete Math. 30, No. 1, 36–42 (2016; Zbl 1329.05244)], our results imply that every sufficiently large \(K_3\)-divisible graph of minimum degree at least \(9 n / 10 + o(n)\) has a \(K_3\)-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large \(K_3\)-divisible graph with minimum degree at least \(3 n / 4\) has a \(K_3\)-decomposition. We also obtain the asymptotically correct minimum degree thresholds of \(2 n / 3 + o(n)\) for the existence of a \(C_4\)-decomposition, and of \(n / 2 + o(n)\) for the existence of a \(C_{2 \ell}\)-decomposition, where \(\ell \geq 3\). Our main contribution is a general ’iterative absorption’ method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams’ conjecture, it suffices to show that every \(K_3\)-divisible graph with minimum degree at least \(3 n / 4 + o(n)\) has an approximate \(K_3\)-decomposition.