  When two solids are squeezed together they will in general not make atomic
contact everywhere within the nominal (or apparent) contact area. This fact has
huge practical implications and must be considered in many technological
applications. In this paper I briefly review basic theories of contact
mechanics. I consider in detail a recently developed contact mechanics theory.
I derive boundary conditions for the stress probability distribution function
for elastic, elastoplastic and adhesive contact between solids and present
numerical results illustrating some aspects of the theory. I analyze contact
problems for very smooth polymer (PMMA) and Pyrex glass surfaces prepared by
cooling liquids of glassy materials from above the glass transition
temperature. I show that the surface roughness which results from the frozen
capillary waves can have a large influence on the contact between the solids.
The analysis suggest a new explanation for puzzling experimental results [L.
Bureau, T. Baumberger and C. Caroli, arXiv:cond-mat/0510232] about the
dependence of the frictional shear stress on the load for contact between a
glassy polymer lens and flat substrates. I discuss the possibility of testing
the theory using numerical methods, e.g., finite element calculations.
