A space with a torus action is called "equivariantly formal" if its equivariant cohomology (say, with rational coefficients) is free over the polynomial ring $H^*(BT)$. Many interesting spaces fall into this class. A nice feature of them is that their equivariant cohomology can be easily computed from the fixed point set and the one-dimensional orbits. It turns out that this last property holds for more spaces than just the equivariant formal ones. We will characterise these spaces and present some examples. We will also discuss equivariant formality for cohomology with integer coefficients.