  We study 1+1 field-generalizations of the rational and elliptic Gaudin
models. For ${\rm sl}(N)$ case we introduce equations of motion and L-A pair
with spectral parameter on the Riemann sphere and elliptic curve. In ${\rm
sl}(2)$ case we study the equations in detail and find the corresponding
Hamiltonian densities. The $n$-site model describes $n$ interacting
Landau-Lifshitz models of magnets. The interaction depends on position of the
sites (marked points on the curve). We also analyze the 2-site case in its own
right and describe its relation to the principal chiral model. We emphasize
that 1+1 version impose a restriction on a choice of flows on the level of the
corresponding 0+1 classical mechanics.
