Abstract: I will discuss two sets of related results concerning critical long-range percolation on Euclidean and hierarchical lattices. For Euclidean lattices, I will show how one can prove power-law bounds on the distribution of the cluster of the origin at criticality, quantifying a theorem of Berger which states that the phase transition is continuous. In the hierarchical case, I will outline forthcoming work giving up-to-constants estimates on point-to-point connection probabilities at criticality and discuss consequences of this for other critical exponents. Both proofs rely crucially on a new rigorous hyperscaling inequality, which I also intend to show the proof of. The part of the talk concerning Euclidean lattices is based on https://arxiv.org/abs/2008.11197.