This paper introduces an object descriptor for classification based on the Euler characteristic of subsets created by thresholding a function at multiple levels (sub-level filtration). We demonstrate the effectiveness of this basic topological invariant of sets, the Euler characteristic, and use it to compute descriptors in two different domains - images and 3D mesh surfaces. The descriptors used as input to linear SVMs achieve state of the art classification results on various public data sets. Moreover, these descriptors are extremely fast to compute. We present linear time methods to calculate the Euler characteristic for multiple threshold values and to compute the Euler characteristic in a sliding window.