************************************************************************* 61125 It has emerged that low luminosity AGN may be much more common than previously thought, accounting for about 10% of galaxies in the local universe (Ho, Filippenko & Sargent 1997). Here we propose ASCA observations of 6 newly discovered local AGN, all with weak broad Halpha, including the least luminous AGN yet known (NGC4395). Combined with existing serendipitous ASCA observations of another 15 local galaxies, this will give us a complete sample of 21 low luminosity AGN to a flux limit of F(Halpha)=10E-13.5 erg sE-1 cmE-2. These observations will allow us to define the very faint end of the AGN X- ray luminosity function and to probe the X-ray characteristic of this extreme end of the AGN phenomenon. We request a 20ks observation for each galaxy, making a total time request of 120ks. *************************************************************************