We demonstrate clear weak anti-localization (WAL) effect arising from induced Rashba spin-orbit coupling (SOC) in WS2-covered single-layer and bilayer graphene devices. Contrary to the uncovered region of a shared single-layer graphene flake, WAL in WS2-covered graphene occurs over a wide range of carrier densities on both electron and hole sides. At high carrier densities, we estimate the Rashba SOC relaxation rate to be ∼ 0.2ps −1 and show that it can be tuned by transverse electric fields. In addition to the Rashba SOC, we also predict the existence of a'valley-Zeeman' SOC from first-principles calculations. The interplay between these two SOC's can open a non-topological but interesting gap in graphene; in particular, zigzag boundaries host four sub-gap edge states protected by time-reversal and crystalline symmetries. The graphene/WS2 system provides a possible platform for these novel edge states.