We have surveyed 188 ROSAT PSPC fields for X-ray sources with hard spectra (alpha<0.5); such sources must be major contributors to the X-ray background at faint fluxes. In this paper we present optical identifications for 62 of these sources: 28 AGN which show broad lines in their optical spectra (BLAGN), 13 narrow emission line galaxies (NELGs), 5 galaxies with no visible emission lines, 8 clusters and 8 Galactic stars. The BLAGN, NELGs and galaxies have similar distributions of X-ray flux and spectra. Their ROSAT spectra are consistent with their being AGN obscured by columns of 20.5<log(NH/cm^2)<23. The hard spectrum BLAGN have a distribution of X-ray to optical ratios which is similar to that found for AGN from soft X-ray surveys (1<alpha_OX<2). However, a relatively large proportion (15%) of the BLAGN, NELGs and galaxies are radio loud. This could be because the radio jets in these objects produce intrinsically hard X-ray emission, or if their hardness is due to absorption, it could be because radio loud objects are more X-ray luminous than radio quiet objects. The 8 hard sources identified as clusters of galaxies are the brightest, and softest group of sources and hence clusters are unlikely to be an important component of the hard, faint population. We propose that BLAGN are likely to constitute a significant fraction of the faint, hard, 0.5 - 2 keV population and could be important to reproducing the shape of the X-ray background, because they are the most numerous type of object in our sample (comprising almost half the identified sources), and because all our high redshift (z>1) identified hard sources have broad lines.
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