We present the task description and discussion on the results of the DCASE 2021 Challenge Task 2. In 2020, we organized an unsupervised anomalous sound detection (ASD) task, identifying whether a given sound was normal or anomalous without anomalous training data. In 2021, we organized an advanced unsupervised ASD task under domain-shift conditions, which focuses on the inevitable problem of the practical use of ASD systems. The main challenge of this task is to detect unknown anomalous sounds where the acoustic characteristics of the training and testing samples are different, i.e., domain-shifted. This problem frequently occurs due to changes in seasons, manufactured products, and/or environmental noise. We received 75 submissions from 26 teams, and several novel approaches have been developed in this challenge. On the basis of the analysis of the evaluation results, we found that there are two types of remarkable approaches that TOP-5 winning teams adopted: 1) ensemble approaches of ``outlier exposure'' (OE)-based detectors and ``inlier modeling'' (IM)-based detectors and 2) approaches based on IM-based detection for features learned in a machine-identification task.