This project aims to develop a theory of pseudo-differential operators and to study the analytical and operator theoretical properties with its applications to partial differential equations on fundamental domains of lattices. This area of periodic pseudo-differential operators is still largely unexplored, even though it would yield valuable results for periodic partial differential equations. To put the construction of this theory in perspective we note that fundamental domains of lattices correspond to linearly deformed tori where we can exploit the pseudo-differential calculus on manifolds. However, this would only lead to a local notion of a symbol, while for many problems in the field of partial differential equations a global symbol is required. While a representation-theoretical approach for obtaining a global quantisation of operators on compact Lie groups has been investigated, these tools are somewhat too abstract and indirect to work with in the setting of periodic pseudo-differential operators. This work will yield concrete analysis based on recent state-of-the-art developments in several closely related fields: pseudo-differential analysis on groups, methods for estimating non-commutative Fourier and spectral multipliers, and their many applications to partial differential equations. As a long term goal this research will be a stepping stone to pseudo-differential theories on more difficult settings in the context of aperiodic order, such as on quasicrystals.