  We introduce a family of spaces called critical varieties. Each critical
variety is a subset of one of positroid varieties in the Grassmannian. The
combinatorics of positroid varieties is captured by the dimer model on a planar
bipartite graph $G$, and the critical variety is obtained by restricting to
Kenyon's critical dimer model associated to a family of isoradial embeddings of
$G$. This model is invariant under square/spider moves on $G$, and we give an
explicit boundary measurement formula for critical varieties which does not
depend on the choice of $G$. This extends our recent results for the critical
Ising model, and simultaneously also includes the case of critical electrical
networks.
  We systematically develop the basic properties of critical varieties. In
particular, we study their real and totally positive parts, the combinatorics
of the associated strand diagrams, and introduce a shift map motivated by the
connection to zonotopal tilings and scattering amplitudes.
