A \emph{covering array} is an $N \times k$ array of elements from a $v$-ary alphabet such that every $N \times t$ subarray contains all $v^t$ tuples from the alphabet of size $t$ at least $λ$ times; this is denoted as $\CA_λ(N; t, k, v)$. Covering arrays have applications in the testing of large-scale complex systems; in systems that are nondeterministic, increasing $λ$ gives greater confidence in the system's correctness. The \emph{covering array number}, $\CAN_λ(t,k,v)$ is the smallest number of rows for which a covering array on the other parameters exists. For general $λ$, only several nontrivial bounds are known, the smallest of which was asymptotically $\log k + λ\log \log k + o(λ)$ when $v, t$ are fixed. Additionally it has been conjectured that the $\log \log k$ term can be removed. First, we affirm the conjecture by deriving an asymptotically optimal bound for $\CAN_λ(t,k,v)$ for general $λ$ and when $v, t$ are constant using the Stein--Lovász--Johnson paradigm. Second, we improve upon the constants of this method using the Lovász local lemma. Third, when $λ=2$, we extend a two-stage paradigm of Sarkar and Colbourn that improves on the general bound and often produces better bounds than even when $λ=1$ of other results. Fourth, we extend this two-stage paradigm further for general $λ$ to obtain an even stronger upper bound, including using graph coloring. And finally, we determine a bound on how large $λ$ can be for when the number of rows is fixed.