  We provide a simple criterion for an element of the mapping class group of a
closed surface to have normal closure equal to the whole mapping class group.
We apply this to show that every nontrivial periodic mapping class that is not
a hyperelliptic involution is a normal generator for the mapping class group
when the genus is at least 3. We also give many examples of pseudo-Anosov
normal generators, answering a question of D. D. Long. In fact we show that
every pseudo-Anosov mapping class with stretch factor less than $\sqrt{2}$ is a
normal generator. Even more, we give pseudo-Anosov normal generators with
arbitrarily large stretch factors and arbitrarily large translation lengths on
the curve graph, disproving a conjecture of Ivanov.
