For hundreds of years mathematicians have been fascinated with slicing high-dimensional mathematical objects as a way to get knowledge and intuition of higher dimensions. There are many classical results and conjectures about slices with hyperplanes (e.g., Bourgain’s conjecture, recently a theorem, on the relation of volume and area of slices). This topic is central to geometry, analysis, and of course combinatorics!! Given a d-dimensional convex polytope P, what is the ``best’’ slice of P by a hyperplane? For a combinatorialists ``best’’ can mean for example, one with the largest number of vertices! Not only we investigate the above combinatorial optimization theorem but also, note that as we slice P with different hyperplanes, we create many combinatorially different (d-1)-slices, which are also polytopes of course. E.g., for a 3-dimensional regular cube there are 4 combinatorial types of slices (triangles, quadrilaterals, pentagons, hexagons). We investigate: How many combinatorially different slices are there for a polytope P? How can we count them all? Can we give lower/upper bounds on their number? What are extremal cases? I will explain a powerful new combinatorial model (a moduli space of slices) and an algorithmic framework that answers these problems (and many others) in polynomial time when dim(P) is fixed. Moreover, we show the problems have hard complexity otherwise. The results are joint work with Marie-Charlotte Brandenburg (U Bochum) and Chiara Meroni (ETH) and Antonio Torres and Gyivan López (UC Davis). This talk will have lots of pretty pictures and will be understandable by everyone. I will present lots of open questions for enthusiastic researchers in the audience.