  An introduction and survey is given of some recent work on the infinitesimal
dynamics of \textit{crystal frameworks}, that is, of translationally periodic
discrete bond-node structures in $\mathbb{R}^d$, for $ d=2,3,...$. We discuss
the rigidity matrix, a fundamental object from finite bar-joint framework
theory, rigidity operators, matrix-function representations and low energy
phonons. These phonons in material crystals, such as quartz and zeolites, are
known as rigid unit modes, or RUMs, and are associated with the relative
motions of rigid units, such as ~SiO$_4$ tetrahedra in the tetrahedral
polyhedral bond-node model for quartz. We also introduce semi-infinite crystal
frameworks, bi-crystal frameworks and associated multi-variable Toeplitz
operators.
