Dynamical systems are mathematical models of change or motion over time. They are ubiquitous in physics, computer science, biology, and many other branches of pure and applied science. In view of several recent developments, there is great opportunity and demand for powerful tools based on mathematical and computational logic for a qualitative formal analysis of these systems. The central question that concerns us is: Question 1.1. Which logical theories are appropriate for describing and rea-soning about dynamical systems that appear in various branches of science and technology? Each logical theory may have several advantages and disadvantages, and the criteria for choosing a suitable framework varies by discipline. Thus we will work both with highly expressive logics stemming from foundations of mathematics, and less expressive but more computationally effective logics stemming fromcomputer science. Specifically, will develop tools for efficient, automated reasoning about continuous dynamical systems applicable in real-world scenarios, such as the modeling of controlled ecosystems for sustainable farming. We will also study frame-works for formal reasoning about transfinite dynamics in discontinuous systems. Such dynamical systems are of interest in proof theory to model Turing progressions, which in turn can be used to produce mathematically natural statements independent from strong formal theories in the spirit of G ̈odel’s incompleteness theorems.