  Low-luminosity AGN (with X-ray luminosity < 1e42 ergs/s) far outnumber
ordinary AGN, and are therefore perhaps more relevant to our understanding of
AGN phenomena and the relationship between AGN and host galaxies. Many normal
galaxies harbor LINER and starburst nuclei, which, together with LLAGN, are a
class of ``low-activity'' galaxies that have a number of surprisingly similar
X-ray characteristics, despite their heterogenous optical classification. This
strongly supports the hypothesis of an AGN-starburst connection. Further, X-ray
observations of normal galaxies without starburst or AGN-like activity in their
nuclei offer opportunities to study populations of X-ray binaries, HII regions,
and warm or hot ISM under different conditions than is often the case in the
Milky Way. The results of recent X-ray observations of these types of galaxies
are reviewed, and what we hope to learn about both nearby and high redshift
galaxies of each type from observations with forthcoming and planned satellites
is discussed.
