Parallel-server queues with customer abandonment model customer call centers and web server farms. Empirical research suggests that service times and customer patience times are not exponentially distributed in general. We focus on queues with a large number of servers to meet the high volume of arrivals. We study many-server fluid and diffusion limits of these queues in two different limit regimes: the overloaded regime and the critically loaded regime; the latter is known as the Halfin-Whitt regime. In both limit regimes, we discuss recent progresses on fluid and diffusion limits. Emphasis is on applying the continuous mapping theorem, which has played a key role in proving conventional heavy traffic limits in queueing networks. The talk is based on joint works with Tolga Tezcan and Shuangchi He.