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We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive p-curvature. The p-curvature was defined and studied by the second author in earlier papers. It turns out that the positivity of p-curvature is preserved under surgeries of codimension at least p + 3. This gives a key to reducing a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connecteddoi:10.1090/s0002-9947-2014-05939-4 fatcat:5qu2az5wjnafpgun7g4r4aywg4