************************************************************************* 73012 We propose observations of 4 nearby galaxies, selected on the basis that their optical nuclear spectra is classified by Ho, Filippenko and Sargent (1997) as a variety of LINER or H II nucleus, a bright counterpart X-ray source is seen within 1' of their nuclear position in the ROSAT All-Sky Survey, and they are previously unobserved by ASCA. The main incentive for studying these nuclei is to expand our knowledge of the X-ray emission from LINERs by observing 3 new examples of this phenomenon. Complementary to this is the search for, and study of, new low-luminosity AGN candidates. This proposal shows that observations with ASCA are an excellent medium for achieving these aims. *************************************************************************