Protoplanetary discs can be deeply affected by the UV radiation produced by massive stars. The process of 'external photoevaporation' could be relevant even in regions with relatively weak UV field. Using a viscous evolution code, and the mass loss rate of the FRIED grid, I simulated the evolution of typical protoplanetary discs subject to various, constant, FUV radiation field strengths. If disc evolution (and accretion) is driven by viscosity, the observed disc sizes should increase with time due to the outward angular momentum transport. Due to external photoevaporation I find that already at a low level of irradiation (10 G0) discs shrink. To quantify how important this is for nearby star forming regions, I calculated the local UV radiation field in Upper Sco, Lupus and Taurus, showing in Upper Sco a median of the UV flux of about 48 G0. In Upper Sco external photoevaporation is the best candidate to explain the small disc sizes detected, because of the high number of B stars, and could play a role in explain the small disc masses (usually associated with stellar evolution). The impact of this process on regions with weak UV radiation field is still to be quantified.