The geometry of secant varieties of Segre varieties is difficult to study, partly because their definition involves taking a closure that is usually poorly understood. To approach this problem, we introduce concise secant varieties, which come equipped with projective birational morphisms to the usual secant varieties and can be viewed as partial desingularizations. A key feature of these new varieties is that their local geometry is closely related to that of well-studied moduli spaces, such as Hilbert schemes of points. We exploit this connection to construct resolutions of singularities of secant varieties in small cases. Moreover, the resulting resolutions admit natural torus actions, allowing us to apply methods of equivariant topology to investigate their finer geometry. We use these results to study the original secant varieties, including applications to the classical Salmon problem. This is joint work with Joachim Jelisiejew.