  This paper, largely written in 2009/2010, fits Landau-Ginzburg models into
the mirror symmetry program pursued by the last author jointly with Mark Gross
since 2001. This point of view transparently brings in tropical disks of Maslov
index 2 via the notion of broken lines, previously introduced in two dimensions
by Mark Gross in his study of mirror symmetry for $\mathbb{P}^2$.
  A major insight is the equivalence of properness of the Landau-Ginzburg
potential with smoothness of the anticanonical divisor on the mirror side. We
obtain proper superpotentials which agree on an open part with those
classically known for toric varieties. Examples include mirror LG models for
non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano
threefolds.
