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To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrakg\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a largecommutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchyof cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in thespecial case $\sigma = \text{id}$.  We go on to construct joint eigenvectors and their eigenvalues for thishierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chainconsisting of a tensor product of Verma modules. To do so we generalize anapproach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involvingvertex algebras and the Wakimoto construction. As part of this construction, wemake use of a theorem concerning cyclotomic coinvariants, which we prove in acompanion paper.  As a byproduct, we obtain a cyclotomic generalization of theSchechtman-Varchenko formula for the weight function.

Cyclotomic Gaudin models: construction and Bethe ansatz