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1) The contribution by Ugo Locatelli focuses on the explicit construction of invariant tori exploiting suitable Hamiltonian normal forms, with particular emphasis on applications to Celestial Mechanics. First, the algorithm constructing the Kolmogorov normal form is described in detail. Then the extension to lowerdimensional elliptic tori is provided. Both algorithms are then combined so as to accurately approximate the long-term dynamics of the HD 4732 extrasolar system. 2) The contribution by Gabriella Pinzari presents a review of some results of their research group, regarding the relation between some particular motions of the Three-Body problem (3BP) and the motions of the so-called Euler (or two-centre) problem, which is integrable. For the analysis of such relation, the authors make use of two novel results: on one hand, the two-centre problem (2CP) bears a remarkable property, here called renormalizable integrability, which states that the simple averaged potential of the 2CP and the Euler integral are one function of the other; on the other hand, the motions of the Euler integral are at least qualitatively explicit, and the averaged Newtonian potential is a prominent part of the 3BP Hamiltonian.3) The contribution by Alessandra Celletti deals with dissipative systems, a key topic in Celestial Mechanics. In particular the problem of the existence of invariant tori for conformally symplectic systems, which have the property to transform the symplectic form into a multiple of itself, is studied. Two different models are presented: a discrete system known as the standard map and a continuous system known as the spin-orbit problem. In both cases, both the conservative and dissipative versions are considered, in order to highlight the differences between the symplectic and conformally symplectic dynamics. 4) The contribution by Gwenael Boue provides basic tools to understand the rotational dynamics of extended bodies which could be either rigid or deformable by tides. The problem is described in a Lagrangian formalism as it was developed by H. Poincare in 1901. The case of rigid body is also presented in the corresponding Hamiltonian formalism. The mathematical description of the deformation of the extended body follows the approach used by C. Ragazzo and L. Ruiz in their two papers of 2015, 2017 due to the compactness and clarity of their formalism. In this Chapter, many applications to the rotation and the libration of celestial bodies are illustrated. 5) The contribution by Christos Efthymiopoulos concerns the phenomenon of Arnold diffusion. The authors begin with the famous example given by Arnold to describe the slow diffusion taking place in the action-space in Hamiltonian nonlinear dynamical systems with three or more degrees of freedom. The text introduces basic concepts related to our current understanding of the mechanisms leading to Arnold diffusion and at the same time performed a qualitative investigation of the phenomenon of Arnold diffusion with many examples. The problem of the speed of diffusion is investigated using methods of perturbation theory, with particular emphasis on Nekhoroshevs theorem. 6) The contribution by Giovanni F. Gronchi deals with the problem of initial orbit determination of a solar system body, i.e. the determination of a preliminary orbit from observations collected for example by a telescope. The two methods that are presented, named Link2 and Link3, try to link together two and three, respectively, short arcs of optical observations of the same object which can possibly be quite far apart in time. The conservation laws of Kepler’s problem are used to derive a polynomial equation of degree 9 (Link2) and 8 (Link3) for the distance of the body from the observer. 7) The contribution by Catalin Gales provides an overview of some recent developments in the study of dynamics of space debris with focus on specific resonant interactions, in particular those related to the tesseral resonances. After an historical introduction to the topic, the authors provide a long-term picture of the dynamics that can help in the modeling and mitigation of the space debris problem, both in term of Cartesian coordinates and in the Hamiltonian framework. Some key terms in the perturbing functions are classified, while the effect of the dissipative force of the atmospheric drag is also formulated. 8) The contribution by Massimiliano Guzzo presents the use of the Fast Lyapunov Indicators (FLI) in the Three-Body problem, with the eventual aim of computing transit orbits. The FLI belong to the family of the finite time indicators, which are able to extract the information of the solutions of the variational equations on short time intervals. First, the FLI are applied to two model problems: the standard map and the double gyre. Then, it is described a modification of the FLI which was originally introduced to improve the computation of stable and unstable manifolds in model systems and the Three-Body problem. 9) The contribution by Antonio Giorgilli provides an answer to a simple question, how did Kepler discover his celebrated laws?. The answer however is not that simple and the present paper guides the reader by a short walk along the main works of Kepler, notably the Astronomia Nova, trying to follow his search of the perfection of the World till the discovery of his celebrated laws. At the end of the road, the consciousness that the finish line had not yet been reached.

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