The talk is focused on the clamped plate problem, initially formulated by Lord Rayleigh in 1877, and solved by M. Ashbaugh & R. Benguria (Duke Math. J., 1995) and N. Nadirashvili (Arch. Ration. Mech. Anal., 1995) in 2 and 3 dimensional euclidean spaces. We consider the same problem on both negatively and positively curved spaces, and provide various answers depending on the curvature, dimension and the width/size of the clamped plate.