There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(Δ+ \log^* n)$ communication rounds; here $n$ is the number of nodes and $Δ$ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on $n$ is optimal: these problems cannot be solved in $o(\log^* n)$ rounds even if $Δ= 2$. However, the dependency on $Δ$ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least $1-1/n$ requires $Ω(\min\{Δ,\log \log n / \log \log \log n\})$ rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires $Ω(\min\{Δ, \log n / \log \log n\})$ rounds; this is an improvement over prior lower bounds also as a function of $n$.