Finiteness results for lattice polytopes

The talk is devoted to the families of lattice polytopes with a fixed number of interior lattice points. It is well-known that such families are important for algebraic geometry. I will present recent results motivated by the work of Hensley, Lagarias, Ziegler, Lawrence and Pikhurko. In particular, I will present new upper bounds on the size of lattice simplices with a single interior lattice point.