Circular spectropolarimetric observations of 74 stars were obtained in an attempt to detect magnetic fields via the longitudinal Zeeman effect in their spectral lines. The sample observed includes 22 normal B, A and F stars, 4 emission line B and A stars, 25 Am stars 10 HgMn stars, 2 lambda Bootis stars and 11 magnetic Ap stars. Using the Least Squares Deconvolution multi-line analysis approach (Donati et al. 1997), Stokes I and V mean signatures were extracted from each spectrum. We find no evidence for magnetic fields in the normal, Am and HgMn stars, with upper limits on longitudinal field measurements usually considerably smaller than any previously obtained for these objects. The implication is that any magnetic fields, if any, which exist in the photospheres of these stars must be considerably weaker and/or considerably more complex than those of active late-type stars. We also make a new detection of a field in the A2pSr star HD 108945 and make new precise measurements of longitudinal fields in other Ap SrCrEu stars. We furthermore report the detection of the spectral lines of the secondary of HD 108642. As well, we report new results for several binary systems, including a new vsini for the rapidly rotating secondary of the Am delta Del SB2 HD 100951.