  The lattice problem for models of Peano Arithmetic ($\mathsf{PA}$) is to
determine which lattices can be represented as lattices of elementary submodels
of a model of $\mathsf{PA}$, or, in greater generality, for a given model
$\mathcal{M}$, which lattices can be represented as interstructure lattices of
elementary submodels $\mathcal{K}$ of an elementary extension $\mathcal{N}$
such that $\mathcal{M} \preccurlyeq \mathcal{K} \preccurlyeq \mathcal{N}$. The
problem has been studied for the last 60 years and the results and their proofs
show an interesting interplay between the model theory of PA, Ramsey style
combinatorics, lattice representation theory, and elementary number theory. We
present a survey of the most important results together with a detailed
analysis of some special cases to explain and motivate a technique developed by
James Schmerl for constructing elementary extensions with prescribed
interstructure lattices. The last section is devoted to a discussion of
lesser-known results about lattices of elementary submodels of countable
recursively saturated models of PA.
