We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for (Formula presented.) in every (Formula presented.) -connected graph any two longest cycles intersect in at least (Formula presented.) vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph, which can be used to improve the best known bounds toward all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on (Formula presented.) vertices contains a cycle (and hence path) of length at least (Formula presented.), improving on (Formula presented.) from DeVos [arXiv:2302:04255, 2023]. Second, we show that in every (Formula presented.) -connected graph with (Formula presented.), any two longest cycles meet in at least (Formula presented.) vertices, improving on (Formula presented.) from Chen, Faudree, and Gould [J. Combin. Theory, Ser. B, 72 (1998) no. 1, 143–149]. Our proof combines combinatorial arguments, computer search, and linear programming.