  We introduce the quasiminimal subshifts, subshifts having only finitely many
subsystems. With $\mathbb{N}$-actions, their theory essentially reduces to the
theory of minimal systems, but with $\mathbb{Z}$-actions, the class is much
larger. We show many examples of such subshifts, and in particular construct a
universal system with only a single proper subsystem, refuting a conjecture of
[Delvenne, K\r{u}rka, Blondel, '05].
