This talk explores some basic issues in the analysis of large spatial datasets when the observation sites are not on a grid. The focus is on situations in which nearby observations tend to have very similar values. Some specific issues I will consider include: why random field models are preferable to regression models, diagnostics for spatial data, and computational challenges in working with large, unstructured covariance matrices. These issues will be illustrated by looking at total column ozone values as measured on a single orbit of the Ozone Monitoring Instrument, yielding 83,305 ungridded observations.