An important goal of EEG research is to obtain practical methods to improve the spatial resolution of scalp-recorded potentials, i.e., to make surface data more accurately represent local underlying brain sources. This goal may be somewhat different from that of "localizing brain activity with EEG," since the latter approach often involves prior assumptions about the nature of sources. In this paper, we demonstrate that spline-Laplacian, a relatively new approach that can yield dramatic improvement in spatial resolution when average electrode spacing is less than about 3 cm. This approach is mostly independent of assumptions about sources and models of the head. The demonstration involves computer simulations, evoked potentials, normal spontaneous EEG, and epileptic spikes.