Anomalies are samples that significantly deviate from the rest of the data and their detection plays a major role in building machine learning models that can be reliably used in applications such as data-driven design and novelty detection. The majority of existing anomaly detection methods either are exclusively developed for (semi) supervised settings, or provide poor performance in unsupervised applications where there is no training data with labeled anomalous samples. To bridge this research gap, we introduce a robust, efficient, and interpretable methodology based on nonlinear manifold learning to detect anomalies in unsupervised settings. The essence of our approach is to learn a low-dimensional and interpretable latent representation (aka manifold) for all the data points such that normal samples are automatically clustered together and hence can be easily and robustly identified. We learn this low-dimensional manifold by designing a learning algorithm that leverages either a latent map Gaussian process (LMGP) or a deep autoencoder (AE). Our LMGP-based approach, in particular, provides a probabilistic perspective on the learning task and is ideal for high-dimensional applications with scarce data. We demonstrate the superior performance of our approach over existing technologies via multiple analytic examples and real-world datasets.