Diophantine problems have always been a central topic in Number Theory, and have shown deep links with other basic mathematical topics, like Algebraic and Complex Geometry. Our research plan focuses on some issues in this realm, which are strictly interrelated. In the last years the PI and collaborators obtained several results on integral and algebraic points on varieties, which have inspired much subsequent research by others, and which we plan to develop further. In particular:We plan a further study of integral points on varieties, and applications to Algebraic Dynamics, a possibility which has emerged recently.We plan to study further the so-called `Unlikely intersections'. This theme contains celebrated issues like the Manin-Mumford conjecture. After work of the PI with Bombieri and Masser in the last 10 years, it has been the object of much recent work and also of new conjectures by R. Pink and B. Zilber. Here a new method has recently emerged in work of the PI with Masser and Pila, which also leads (as shown by Pila) to signi_cant new cases of the Andr_e-Oort conjecture. We intend to pursue in this kind of investigation, exploring further the range of the methods.Finally, we plan further study of topics of Diophantine Approximation and Hilbert Irreducibility, connected with the above ones in the contents and in the methodology.