This thesis reports on the study of a volume-linked sample of 46 galaxies. It forms part of a continuing, multiwavelength program to assess the prevalence of low luminosity nuclear activity in galaxies with a wide range of sizes. X-ray observations are used, together with images at other wavelengths from previous work, to create a list of possible sites of nuclear activity. Optical spectra of these targets are analysed. The X-ray and optical properties of the nuclear regions, and their relationship with the host galaxies, are investigated. Finally, one particular object, NGC 4395, is described in detail. ROSAT X-ray images were studied for 29 galaxies. Contour maps are presented for all galaxies with detected emission and fluxes are given for all observed point sources. Upper limits are given for galaxies with no detected nuclear sources. In general, X-ray nuclear sources are found to be extremely common. Their fluxes correlate strongly with the host galaxy luminosity (the analysis includes results from the literature for another 4 galaxies). Long slit, high resolution, ˜ 3600 - 6800 Å optical spectra were obtained for all the nuclear targets identified after careful examination of radio, IR, optical and X-ray images. Spectra are presented for each target and emission line fluxes are measured. Emission line diagnostic diagrams are used to investigate the nature of the ionising sources of the line-emitting targets. Results are compared with the spectral classification given by Ho, Filippenko and Sargent (1997, Ap. J. Suppl., 112, 315) for galaxies common to both surveys and objects without previous observations are classified. The fraction of optically identified AGN in the sample is obtained. Based on the observed correlation between X-ray and host galaxy luminosity, the properties of the nuclear X-ray sources are investigated. Nuclear Hα and X-ray luminosities are compared with relationships established for more luminous AGN and with predictions from starburst models. Correlations are also found between nuclear Hα luminosities and host absolute magnitudes for nuclei optically classified as starbursts and AGN. Tentative explanations for the observed correlations are explored. As part of the multiwavelength project, optical spectra and X-ray images of NGC 4395, the least luminous known Seyfert 1, were obtained. The ROSAT observations show a change in the X-ray flux by a factor of ˜ 2 in 15 days. The spectra show a change in both the luminosity and the shape of the continuum, becoming bluer when brighter. A power law fit to the observations shows that the spectral shape changes between a low state (with α ˜ 2), a medium state (α ˜ 2), a medium state (α ˜ 1), and a high state (α ˜ 0). A week of ground-based optical broad band monitoring of NGC 4395 has also given evidence of a variation in the flux by ˜ 20% in less than 24 hours.