Most commonly used Deep Learning libraries nowadays.
Keras is the best in this category: usable, powerful and actively developed. It can use Tensorflow, Theano, and CNTK as a backend.
TensorFlow from Google (C++/Python). It is built on the top of keras.
CNTK from Microsoft (training in Python & evaluation in C++/C#/Java/Python)
Caffe from Berkeley Vision and Learning Center in C++ with Python bindings
PyTorch from Facebook, in Python, can be extended with C/C++
Hope this answer helps.