  We study the boundary behavior of rational inner functions (RIFs) in
dimensions three and higher from both analytic and geometric viewpoints. On the
analytic side, we use the critical integrability of the derivative of a
rational inner function of several variables to quantify the behavior of a RIF
near its singularities, and on the geometric side we show that the unimodular
level sets of a RIF convey information about its set of singularities. We then
specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study
of their derivative integrability, zero set and unimodular level set behavior,
and non-tangential boundary values. Our results, coupled with constructions of
non-trivial RIF examples, demonstrate that much of the nice behavior seen in
the two-variable case is lost in higher dimensions.
