Graphene-polymer hydrogels and aerogels hold interest for both fundamental studies of graphene-polymer interactions as well as the production of nano-enhanced polymeric materials. Flexible, compressible, and self-healing pristine graphene/polymer hydrogels were synthesized via in-situ polymerization of the monomer in the polymer-stabilized graphene dispersion. The graphene sheets act as physical cross-linkers and permit gelation without the presence of any chemical cross-linker. Rheological measurements indicate that these physically cross-linked gels have higher storage modulus and toughness compared to the chemically cross-linked baseline. These gels are turned into conductive aerogels (or cryogels) by critical point drying or freeze drying. The aim is to create percolating composites with ultralow filler content by utilizing aerogels or cryogels as a conductive template. This is done by backfilling and polymerizing epoxy resin into the scaffold. The infusion of the resin does not disrupt the monolithic structure or conductive network. Three different aerogel systems (inorganic, organic and polymer aerogel) are investigated with both graphene and carbon nanotubes loading, with successful percolation in each case. A percolation threshold as low as 0.012 vol. % is obtained for graphene loaded organic aerogel/epoxy composite. This is the lowest reported threshold of any graphene based nanocomposites.