Surfaces that refract, but do not reflect I know there exists surfaces that only reflect light (mirrors), but are there surfaces that only refract light? If so, how does that happen?
 A: The Fresnel Equations describes the portion of the electromagnetic field that is reflected at a surface. For any indefinite and non-engineered material, in order to have refraction you must have an index greater than 1, and thus you must have reflection at the surface. There are two significant exceptions to this.
First, if you create a slab of material such that the reflected wave off of the second boundary is both perfectly out of phase with the wave reflected off of the first surface AND of equal amplitude, the reflected waves will destructively interfere causing no reflection, but transmission will still occur. This will lead to refraction but no perceived reflection.
Secondly, if you were to engineer an isotropic metamaterial such that it had an index of -1, the surface will have no index mismatch so there will be no reflected wave, but "refraction" will still occur. This is only a theoretical condition, as no isotropic negative index metamaterial has been fabricated. For a brief overview of metamaterials see here.
A: Besides the slab structure (not a material) and negative index material (quite hard to obtain, even in a lab), I would like to add another category of materials with the desired properties. For a material with anisotropic permittivity, the simplest example being uniaxial crystal, you have pretty chance to have only refraction. This is actually the concept of uniaxial perfectly matched layer (UPML). See (Ref. 4 in) http://en.wikipedia.org/wiki/Perfectly_matched_layer for more details. 
