Abstract: No-k-equal spaces are the spaces of (ordered) points with no k of them occupying the same position in a topological space X (for k=2 they are, of course, the "usual" configuration spaces). They became popular as a testing ground for lower bounds in TCS, but are interesting on their own right (if X is a vector space, the no-k-equal spaces are subspace arrangement manifolds, with a lot of machinery to understand them). In this talk I will survey what we know about the Betti numbers of these configuration spaces in simplest situations. In particular, we'll see that the exponential generating function for the Poincare polynomials for the no-k-equal spaces on the interval is \[ \sum_n P_n(t)\frac{z^n}{n!} =\frac{e^z}{1 +(-t)^{k-2}(e^z q_k(z) - 1)}. \] Some problems will be posed.