The discovery of two-dimensional (2D) magnetic materials with high critical temperature and intrinsic magnetic properties has attracted significant research interest. By using swarm-intelligence structure search and first-principles calculations, we predict three 2D iron arsenide monolayers (denoted as FeAs-I, II and III) with good energetic and dynamical stabilities. We find that FeAs-I and II are ferromagnets, while FeAs-III is an antiferromagnet. FeAs-I and III have sizable magnetic anisotropy comparable to the magnetic recording materials such as the FeCo alloy. Importantly, we show that FeAs-I and III have critical temperatures of 645 K and 350 K, respectively, which are above room temperature. In addition, FeAs-I and II are metallic, while FeAs-III is semiconducting with a gap comparable to Si. For FeAs-III, there exist two pairs of 2D antiferromagnetic Dirac points below the Fermi level, and it displays a giant magneto band-structure effect. The superior magnetic and electronic properties of the FeAs monolayers make them promising candidates for spintronics applications.