We use the genus theory to prove the existence and multiplicity of solutions for the fractional p-Kirchhoff problem − [ M ( ∫ Q |u(x) − u(y)| p |x − y|N+ps dx dy) ]p−1 (−∆)s pu = λh(x, u) in Ω, u = 0 on ℝ N \ Ω, where Ω is an open bounded smooth domain of ℝ N , p > 1, N > ps with s ∈ (0, 1) fixed, Q = ℝ 2N \ (CΩ × CΩ), λ > 0 is a numerical parameter, M and h are continuous functions.