  We explore the stability of three-dimensional Weyl and Dirac semimetals
subject to quasiperiodic potentials. We present numerical evidence that the
semimetal is stable for weak quasiperiodic potentials, despite being unstable
for weak random potentials. As the quasiperiodic potential strength increases,
the semimetal transitions to a metal, then to an "inverted" semimetal, and then
finally to a metal again. The semimetal and metal are distinguished by the
density of states at the Weyl point, as well as by level statistics, transport,
and the momentum-space structure of eigenstates near the Weyl point. The
critical properties of the transitions in quasiperiodic systems differ from
those in random systems: we do not find a clear critical scaling regime in
energy; instead, at the quasiperiodic transitions, the density of states
appears to jump abruptly (and discontinuously to within our resolution).
