Keras - high level nn API that runs on top of TensorFlow or Theano. A good place to start.
TensorFlow - library for distributed numerical computation and data flow graphs.
Theano - popular Python library.
mxnet - fast and flexible. Originally from University of Washington and Carnegie Mellon University adopted by AWS and now an Apache incubator project.
Microsoft Cognitive Toolkit (CNTK) - Microsoft's Cognitive Toolkit.
neon Intel/Nervana's framework.
DeepLearning4J - commercial grade library for Java.
Cortex - new library for Clojure from ThinkTopic and Mike Anderson.
H20 - interesting application with many different algorithms.
Torch - popular libary based on Lua.
PyTorch - Python based framework from Facebook based on Torch.
Caffe - popular library for images and CNNs.
Natural Language Toolkit - the leading platform for NLP education and exploration.
spaCy - newer more production oriented, minimal and opinionated.
gensim - topic modelling for humans.
Deep Learning / Goodfellow, Bengio and Courville - text book by top DL scientists.
Deep Learning with Python / Chollet - from the author of Keras.
Deep Learning: A Practitioner's Approach / Patterson and Gibson - from the authors DeepLearning4j.