We discuss recent progress concerning the resummation of large logarithms at next-to-leading power (NLP) in scattering processes such as Drell-Yan and deep inelastic scattering near threshold, and thrust in the two-jet limit. We start by reviewing the approach based on soft-collinear effective field theory and show that the standard factorization into short distance coefficients, collinear and soft functions at NLP leads in general to the appearance of endpoint divergences, which prevent the naive application of resummation techniques based on the renormalization group. Taking thrust as a case study, we then show that these singularities are indeed an artifact of the effective theory, and discuss how they can be removed to recover a finite factorization theorem and achieve resummation at NLP, at LL accuracy. Last, we discuss recent work concerning the calculation of all collinear and soft functions necessary to reproduce Drell-Yan near threshold up to NNLO in perturbation theory. This calculation provides useful data to extend resummation at NLP beyond LL accuracy.