A cutting element for use with an earth-boring drill bit includes a diamond cutting table that is substantially free of a metallic binder. The cutting table may include polycrystalline diamond and a carbonate binder or polycrystalline diamond with silicon and/or silicon carbide dispersed therethrough. A base of the cutting table is secured to a substrate by way of an adhesion layer. The adhesion layer includes diamond. The adhesion layer may also include cobalt or another suitable binder material, which may be mixed with diamond particles from which the adhesion layer is formed, or may leach from the substrate into the adhesion layer as the cutting element is bonded to the substrate. Alternatively, the cutting table may be formed from and consist essentially of chemical vapor deposited diamond that has been diamond bonded to an underlying polycrystalline diamond compact. Processes may include securing substantially metallic binder-free cutting elements to substrates.