We introduce the notion of critical toric surfaces, with a focus on the so-call "critical toric blow-ups".  These are symplectic manifolds with a toric action on the complement of an isotropic Lagrangian skeleton.  We explain some applications of this construction on new constructions of finite group actions; understanding the symplectic mapping class groups; and the construction of Lagrangian torus fibrations.

This is joint work with Liat Kessler.