Despite more than 30 years of searches, the compact object in Supernova (SN) 1987A has not yet been detected. We present new limits on the compact object in SN 1987A using millimeter, near-infrared, optical, ultraviolet, and X-ray observations from ALMA, VLT, HST, and Chandra. The limits are approximately 0.1 mJy ($0.1\times 10^{-26}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$) at 213 GHz, 1 Lsun ($6\times 10^{-29}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$) in optical if our line-of-sight is free of ejecta dust, and $10^{36}$ erg s$^{-1}$ ($2\times 10^{-30}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$) in 2-10 keV X-rays. Our X-ray limits are an order of magnitude less constraining than previous limits because we use a more realistic ejecta absorption model based on three-dimensional neutrino-driven SN explosion models (presented in an accompanying article). The allowed bolometric luminosity of the compact object is 22 Lsun if our line-of-sight is free of ejecta dust, or 138 Lsun if dust-obscured. Depending on assumptions, these values limit the effective temperature of a neutron star to <4-8 MK and do not exclude models, which typically are in the range 3-4 MK. For the simplest accretion model, the accretion rate for an efficiency $\eta$ is limited to $< 10^{-11} \eta^{-1}$ Msun yr$^{-1}$, which excludes most predictions. For pulsar activity modeled by a rotating magnetic dipole in vacuum, the limit on the magnetic field strength ($B$) for a given spin period ($P$) is $B < 10^{14} P^2$ G s$^{-2}$. By combining information about radiation reprocessing and geometry, it is likely that the compact object is a dust-obscured thermally-emitting neutron star, which may appear as a region of higher-temperature ejecta dust emission.
Comment: 35 pages, 10 figures. Accepted for publication in ApJ. Updated to match accepted version; slightly shortened and clarified the discussion