Equations of prescribed k-th mean curvature do not have
complete interior regularity, in the sense that solutions
may be nonsmooth even if the right hand side of the equation
is smooth. Some additional information is needed to derive
interior estimates for the second derivatives. The results
proved so far require either that the second derivatives are
integrable to a sufficiently high power, or else require some
condition on the boundary data. I will describe these results.