We present the results of ASCA observations of a heterogenous sample of 15 spiral galaxies . 8 are LINERs or low-luminosity AGN ( LLAGN ) , 5 are starburst galaxies and 2 are normal spiral galaxies . We find that in all cases the ASCA spectra can be described by a canonical model consisting of a power-law with a photon index , \Gamma \sim 1.7 - 2.0 , plus a soft optically thin emission component with kT \sim 0.6 - 0.8 keV . The implied element abundances are often sub-solar . The soft component is usually extended and the nuclear , point-like emission is sometimes absorbed by column densities in the range \sim 10 ^ { 21 } -10 ^ { 23 } cm ^ { -2 } . The relative luminosities of the soft and hard components vary from galaxy to galaxy . For the LINERs , the 2-10 keV luminosity of the hard component is typically \sim 10 ^ { 40 - 41 } ergs s ^ { -1 } whereas the 0.5-2.0 keV luminosity of the soft component is typically \sim 10 ^ { 39 - 40 } ergs s ^ { -1 } . For starbursts , the 2-10 keV luminosity of is \sim 10 ^ { 39 - 40 } ergs s ^ { -1 } , somewhat lower than the corresponding luminosity of most of the LINERs in our sample . The hard component is similar to the observed X-ray spectra of quasars and also to the intrinsic X-ray spectra of classical Seyfert galaxies . Most of the galaxies in our sample exhibit no significant ( \Delta I / I > 20 \% ) short-term variability ( with timescales of a day or less ) whereas long-term variability is common . We present a case study of the LINER M81 in detail where there is evidence of large-amplitude ( \Delta I / I \sim 70 \% ) variability over several weeks . There is also clear evidence for a broad , complex Fe-K emission line which is compatible with an origin in an accretion disk viewed at \sim 40 degrees . These results suggest a strong connection between classical AGN , LINERs , and starburst galaxies .