  This paper adopts a tool from computational topology, the Euler
characteristic curve (ECC) of a sample, to perform one- and two-sample goodness
of fit tests. We call our procedure TopoTests. The presented tests work for
samples of arbitrary dimension, having comparable power to the state-of-the-art
tests in the one-dimensional case. It is demonstrated that the type I error of
TopoTests can be controlled and their type II error vanishes exponentially with
increasing sample size. Extensive numerical simulations of TopoTests are
conducted to demonstrate their power for samples of various sizes.
