Scientists explore the ultimate matter components By probing it with photons and electrons. Targeting this goal, I present novel theoretical and computational techniques for exploring atoms and ions by means of light scattering and laser assisted electron scattering. These scenarios are nowadays feasible thanks to new generation of light sources like synchronous and free electron lasers. This book guides the reader step by step to master how electron-matter and photon-matter interactions are described in the language of relativistic as well as non-relativistic quantum mechanics. Special attention is given to Rayleigh scattering, derivation of atomic form factors and form factor shifts brought by laser assistance in electron scattering. This book presents new horizons aimed at discovering matter using cutting-edge fundamental theories.