The major afferent cortical pathway in the visual system passes through the dorsal lateral geniculate nucleus (LGN), where nerve signals originating in the eye can first interact with brain circuits regulating visual processing, vigilance, and attention. Here we asked how on-going and visually driven activity in magnocellular (M), parvocellular (P), and koniocellular (K) layers of the LGN are related to cortical state. We recorded extracellular spiking activity in the LGN simultaneously with local field potentials (LFP) in primary visual cortex, in sufentanil-anesthetized marmoset monkeys. We found that asynchronous cortical states (marked by low power in delta-band LFPs) are linked to high spike rates in K cells (but not P cells or M cells), on multi-second timescales. Cortical asynchrony precedes the increases in K cell spike rates by 1-3 s, implying causality. At sub-second timescales, the spiking activity in many cells of all (M, P, and K) classes is phase-locked to delta waves in the cortical LFP, and more cells are phase-locked during synchronous cortical states than during asynchronous cortical states. The switch from low-to-high spike rates in K cells does not degrade their visual signalling capacity. To the contrary, during asynchronous cortical states the fidelity of visual signals transmitted by K cells is improved, likely because K cell responses become less rectified. Overall the data show that slow fluctuations in cortical state are selectively linked to K pathway spiking activity, whereas delta-frequency cortical oscillations entrain spiking activity throughout the entire LGN, in anaesthetised marmosets. This article is protected by copyright. All rights reserved