The detection of gravitational waves with LIGO and Virgo requires a detailed understanding of the response of these instruments in the presence of environmental and instrumental noise. Of particular interest is the study of anomalous non-Gaussian noise transients known as glitches, since their high occurrence rate in LIGO/Virgo data can obscure or even mimic true gravitational wave signals. Therefore, successfully identifying and excising glitches is of utmost importance to detect and characterize gravitational waves. In this article, we present the first application of Deep Learning combined with Transfer Learning for glitch classification, using real data from LIGO's first discovery campaign labeled by Gravity Spy, showing that knowledge from pre-trained models for real-world object recognition can be transferred for classifying spectrograms of glitches. We demonstrate that this method enables the optimal use of very deep convolutional neural networks for glitch classification given small unbalanced training datasets, significantly reduces the training time, and achieves state-of-the-art accuracy above 98.8%. Once trained via transfer learning, we show that the networks can be truncated and used as feature extractors for unsupervised clustering to automatically group together new classes of glitches and anomalies. This novel capability is of critical importance to identify and remove new types of glitches which will occur as the LIGO/Virgo detectors gradually attain design sensitivity.