The focus of the lecture is a topic to which Prof. Noga Alon devoted much efforts over the years. It is still the subject of some of his present work and of the research of a considerable number of additional researchers. This is the Combinatorial Nullstellensatz, a variant of Hilbert's Nullstellensatz which is one of the foundational results of Algebraic Geometry. This variant found applications in Combinatorics, Graph theory, Additive Number Theory and Combinatorial Geometry. He will describe some of these applications and discuss briefly their intriguing algorithmic aspects.