An Euler structure is a first order structure with a function (an abstract Euler characteristic) assigning elements of a ring to all definable sets, such that several basic properties of counting are satisfied. This notion is interesting on its own, but it is also linked to solvability of uniform (definable) families of linear equations and degrees of Nullstellensatz expressions. I shall give definitions, examples and a few basic facts, and an application to proof complexity, and discuss open problems. A copy of the paper is available on the Web at: http://www.math.cas.cz/~krajicek/euler.ps.gz