In this work, we lay the foundations of UQ for spatially-extended neurobiological models, using neural field equations as a prototype of large-scale cortical activity. More specifically, we cast the equations as a Cauchy problem on abstract Banach spaces subject to random data. We study how the model solution propagates uncertainty in the synaptic kernel, firing rate, initial condition, and external stimulus. Guided by the existing literature on PDEs with random data, we demonstrate the existence of regular solutions in suitable Bochner spaces. We also provide the well- posedness and regularity results for semi-discrete ver- sions of neural fields, in abstract form, and for generic spatial projectors. Moreover, we present a rigorous analysis and numerical results of stochastic collocation schemes for these problems.