In a previous work we have detailed the requirements for obtaining maximal deep learning performance benefit by implementing fully connected deep neural networks (DNN) in the form of arrays of resistive devices. Here we extend the concept of Resistive Processing Unit (RPU) devices to convolutional neural networks (CNNs). We show how to map the convolutional layers to fully connected RPU arrays such that the parallelism of the hardware can be fully utilized in all three cycles of the backpropagation algorithm. We find that the noise and bound limitations imposed by the analog nature of the computations performed on the arrays significantly affect the training accuracy of the CNNs. Noise and bound management techniques are presented that mitigate these problems without introducing any additional complexity in the analog circuits and that can be addressed by the digital circuits. In addition, we discuss digitally programmable update management and device variability reduction techniques that can be used selectively for some of the layers in a CNN. We show that a combination of all those techniques enables a successful application of the RPU concept for training CNNs. The techniques discussed here are more general and can be applied beyond CNN architectures and therefore enables applicability of the RPU approach to a large class of neural network architectures.