  We study the spectrum of two kinds of operators involving a conical geometry:
the Dirichlet Laplacian in conical layers and Schr\"odinger operators with
attractive $\delta$-interactions supported by infinite cones. Under the
assumption that the cones have smooth cross-sections, we prove that such
operators have infinitely many eigenvalues accumulating below the threshold of
the essential spectrum and we express the accumulation rate in terms of the
eigenvalues of an auxiliary one-dimensional operator with a curvature-induced
potential.
