This is an interdisciplinary course including the fundamental laws of the mechanics and physics of crystalline solids, the general description of a periodic structure and their specific characterization methods. We will startwith tensor algebra including definitions, presentations and operations among scalars, vectors, and higher order tensors. Mostly, we will use direct notation and derivatives with respect to the Rectangular Cartesian Coordinates. We will introduce the kinematic of deformation for a continuous body: definitions of deformation mapping, deformation gradients, displacement gradient, change of length and change of area during deformation and motion. After that, we will discuss the material time derivative, the incompressibility, the Lagrangian description and the Euler description followed by the conservation and balance laws. Then, we will give a underlying mathematical framework for the description of periodic structures, i.e. crystalline solids. This part includes the definition of Bravais lattice in 2D and 3D, reciprocal lattice, point group and space group. We will show how the lattice symmetry operation and laws of mechanics are related, and derive a mathematicalmodel for the determination of transformation strain between crystalline solids of different symmetries. Finally, we will introduce the X-ray Laue diffraction and X-ray powder diffraction for the determination of the structural parameters of the crystalline solids, the data analysis and basic indexing procedure and phase determination.