We first characterize all faithful arc-transitive covers of complete graphs and we give a general construction of such covers. For a counterpart of cover, we say a graph $Γ$ is a pseudocover of its quotient $Σ$ if they have the same valency and $Γ$ is not a cover of $Σ$.As the second result of this paper, we prove that the complete graph $\K_n$ has a connected arc-transitive pseudocover if and only if $n-1$ is not a prime.