We study the $F$-decomposition threshold $δ_F$ for a given graph $F$. Here an $F$-decomposition of a graph $G$ is a collection of edge-disjoint copies of $F$ in $G$ which together cover every edge of $G$. (Such an $F$-decomposition can only exist if $G$ is $F$-divisible, i.e. if $e(F)\mid e(G)$ and each vertex degree of $G$ can be expressed as a linear combination of the vertex degrees of $F$.) The $F$-decomposition threshold $δ_F$ is the smallest value ensuring that an $F$-divisible graph $G$ on $n$ vertices with $δ(G)\ge(δ_F+o(1))n$ has an $F$-decomposition. Our main results imply the following for a given graph $F$, where $δ_F^\ast$ is the fractional version of $δ_F$ and $χ:=χ(F)$: (i) $δ_F\le \max\{δ_F^\ast,1-1/(χ+1)\}$; (ii) if $χ\ge 5$, then $δ_F\in\{δ_F^{\ast},1-1/χ,1-1/(χ+1)\}$; (iii) we determine $δ_F$ if $F$ is bipartite. In particular, (i) implies that $δ_{K_r}=δ^\ast_{K_r}$. Our proof involves further developments of the recent `iterative' absorbing approach.