IRAS spatial observations of M33 are presented and compared to ultraviolet, optical, and radio wavelength data. In the infrared the emission structure of M33 appears as a diffuse disk with two bright spiral arms. Prominent localized sources of emission are at the position of the galaxy nucleus and brightest H II region complexes. At 60 and 100 μm the emission extends as far as the UV, the optical, and the H I emission. The global infrared properties of M33 are characteristic of similar quiescent spirals. The far-infrared (40-120 μm) flux is ∼25% of the emission at optical wavelengths, and the total infrared (5 μm-1 mm) emission of dust, estimated using a bolometric correction to the IRAS measurements, is ∼20% of the combined integrated stellar light observed in the UV, optical, and near-infrared. The global and spatial properties of the M33 mid-infrared (8-30 μm) emission indicate that this emission, like the far-infrared emission, originates from dust in the interstellar medium. The radial distributions of the infrared brightness decrease exponentially with radius. The scale length of the 12 μm emission is significantly smaller than the values computed for the UV, visible, and near-infrared emission, and the value derived for the production rate of Lyman continuum photons. The 100 μm scale length, while larger than the 12 μm scale length, is still smaller than the UV and visible light values. The radial distributions of the Iν (12 μm)/Iν (25 μm), Iν (25 μm),/Iν (60 μm), and Iν (60 μm)/Iν (100 μm) infrared colors are flat or decrease only slightly with radius implying near constant dust temperatures across the disk of the galaxy. While the integrated far-infrared and radio continuum fluxes of M33 follow the relationship seen in other galaxies, the radial scale length of the infrared emission is smaller than that of the radio emission. A comparison of the spatial distributions of the far-infrared and radio continuum emission within M33 indicates that the infrared emission is more concentrated around regions of star formation than the nonthermal component of the radio emission. The infrared emission of 19 IRAS point sources detected in M33 are reported; all of these sources coincide with H II regions cataloged at optical and radio wavelengths. Based on the bolometric infrared luminosity of the best detected sources (∼2-70 × 106 L⊙), these regions are similar to the largest star-forming regions in our Galaxy. High-resolution maps of the center and southern arm of the galaxy show a striking correlation between the spatial structure of the infrared emission and the distribution of H II regions. The average infrared properties of detected H II regions are computed, and by comparing the infrared excess of the mean H II region emission to the global infrared excess of the galaxy, we estimate that H II regions account for about half of the integrated infrared emission of M33. The observational results are interpreted using a simplified model of dust heating by the general interstellar radiation field of M33. The model calculations are in good agreement with the observational data when the dust-to-gas abundance is decreased by a factor of 3 from the galaxy center to the outer parts. The model also shows that the UV photons of young B and A stars are the dominant heating source of dust associated with the diffuse ISM. Taking into account the substantial amount of emission from dust associated with high-mass star-forming regions, we find that ∼75% of the integrated infrared emission of M33 is powered by stars younger than a few times 108 yr.