We present observations of the Orion Nebula made with the Hubble Space Telescope in which a number of stars are seen surrounded by dark silhouettes seen projected against the bright background HII region. We find a variety of morphologies, all consistent with thin circumstellar disks spanning a range of diameters (50 to 1000 AU) and inclination angles (0 to > 80 degrees). The silhouette intensity profiles cannot be fit by standard disk models in which the surface density follows a radial power law with an exponent in the range −0.75 to −1.5. Rather, the data are best fit by opaque inner disks with exponential edges, and we discuss possible physical origins of this apparent truncation. Masses in the range 6 × l026−4 × 1030 g (i.e., up to 0.002 M
⊙) are determined for the disks by assuming that the faint light measured from them is background light transmitted through the disk. However, these are strict lower limits on the true disk masses, as most of this light can be accounted for by PSF blurring and scattering in the HST optical train; the present observations are in fact consistent with completely opaque disks. Central stars are seen directly in five of the silhouettes, while the presence of a star is inferred in the sixth, where small reflection nebulae are seen above and below the plane of the near edge-on disk. Optical and near-infrared stellar photometry is consistent with young (~ 1 Myr) low-mass (0.3−1.5 M
⊙) stars, with several showing evidence for excess near-infrared emission from the disk inner edge. These direct imaging observations are discussed in the wider context of circumstellar disks in the Orion Nebula and Trapezium Cluster, additionally revealed as compact ionized nebulae (so-called “proplyds”) in the vicinity of the central OB stars, and via infrared (>2µm) excesses in stellar photometry. Overall, disks are found to be common in the cluster (>50% of all stars), implying that they can survive the rigours of life near massive stars.