Using infrared and X-ray diagnostics, we study star--formation and black hole accretion in nearby and distant galaxies. We examine diagnostics of the hardness of the ionizing field in low--redshift starburst galaxies, to constrain the initial mass function. We obtain new measurements of HeI 1.7 micron/Br 10, a physically simple diagnostic, then test ISO mid--infrared line ratios, finding them reliable. Compared to new photoionization models, the ISO ratios in 27 nearby starburst galaxies are systematically low. This argues that solar--metallicity starbursts are deficient in massive stars, or that such stars are present but highly embedded. Using Spitzer, HST, Chandra, and ground-based data, we examine the multi-wavelength (0.4--24 micron) spectral energy distributions and X-ray properties of X-ray--selected active galactic nuclei (AGN) in several deep fields: the Chandra Deep Field South, the Lockman Hole, and the extended Groth Strip. We examine the 24 micron to X-ray flux and luminosity ratios for 157 AGN at z1̃; the luminosity ratios have not strongly evolved since z0̃, and we find no trend with X-ray column density. This means that highly--obscured AGN do not have exceptional infrared fluxes. We examine the SEDs of 45 bright X-ray and 24 micron sources: only 22% are classified as unobscured t̀̀ype 1'' AGN; 18% are classified as ULIRG-like SEDs; and the majority are classified as obscured (t̀̀ype 2'') AGN or spiral--like SEDs. This supports the picture from X-ray surveys that much of the AGN activity in the distant universe is significantly obscured. We examine why 20% of X-ray--selected AGN are optically--faint; they lie at significantly higher redshifts (median z=1.6) than most X-ray--selected AGN, and their spectra are intrinsically red. Their contribution to the X-ray Seyfert luminosity function is comparable to that of optically--bright AGN at z>1, but they do not significantly alter the redshift distribution. Lastly, we investigate why half of X-ray--selected AGN lack signs of accretion in optical spectra. We find that these ò̀ptically--dull'' AGN have Seyfert--like mid--infrared emission, which argues that they do not have abnormally--weak UV/optical continua. The axis ratios of their host galaxies argue that extinction by host galaxies plays a key role in hiding nuclear emission lines.