Mercer's celebrated theorem is refined and extended by introducing a novel class of higher-order kernel operators that includes the common integral operator only as a special case. \[ \] These operators genuinely take into account information encoded in the (weak) derivatives of a kernel, and their natural domains are Sobolev spaces of order k over some bounded d-dimensional space. domain, where k depends on the order of (weak) differentiability. \[ \] The spectral decomposition of such higher-order kernel operators leads to Mercer-type expansions, which are optimal in terms of the Sobolev norm and, if k>d, also converge uniformly without requiring the kernel to be positive definite. \[ \] Nuclearity of higher order kernel operators is confirmed for positive definite kernels, and a major refinement of Mercer's theorem is obtained that implies trace formulas and a simple rate for the uniform convergence (including derivatives) in terms of the eigenvalues. A further immediate consequence is novel spectral representations of RKHS's. \[ \] Finally, applied to the covariance kernel of a (weakly) differentiable stochastic process, these refinements also yield novel Karhunen-Loève-type expansions allowing for simultaneous approximations of the process and its (weak) derivatives in a mean-square-optimal sense.