This work focuses on several models of aperiodic order and their applications in different areas of mathematics and mathematical physics. After a short introduction to the theory of dynamicals systems, we present random substitutions as a stochastic variant of substitutions which create symbolic dynamical systems that combine long-range order with a positive entropy. Using renormalization techniques, we obtain expressions for the entropy, diffraction, and ergodic measures of such systems. In the second part, we investigate spectral properties of Schr��dinger operators that are associated with non-primitive substitution systems and dynamically defined product systems. Finally we perform a multifractal analysis of a particular spectral measure that has become known as the Thue--Morse measure.