We study network pruning which aims to remove redundant channels/kernels and hence speed up the inference of deep networks. Existing pruning methods either train from scratch with sparsity constraints or minimize the reconstruction error between the feature maps of the pre-trained models and the compressed ones. Both strategies suffer from some limitations: the former kind is computationally expensive and difficult to converge, while the latter kind optimizes the reconstruction error but ignores the discriminative power of channels. In this paper, we propose a simple-yet-effective method called discriminationaware channel pruning (DCP) to choose the channels that actually contribute to the discriminative power. To this end, we first introduce additional discrimination-aware losses into the network to increase the discriminative power of the intermediate layers. Then, we select the most discriminative channels for each layer by considering the discrimination-aware loss and the reconstruction error, simultaneously. We then formulate channel pruning as a sparsity-inducing optimization problem with a convex objective and propose a greedy algorithm to solve the resultant problem. Note that a channel (3D tensor) often consists of a set of kernels (each with a 2D matrix). Besides the redundancy in channels, some kernels in a channel may also be redundant and fail to contribute to the discriminative power of the network, resulting in kernel level redundancy. To solve this, we propose a discrimination-aware kernel pruning (DKP) method to further compress deep networks by removing redundant kernels. To prevent DCP/DKP from selecting redundant channels/kernels, we propose a new adaptive stopping condition, which helps to automatically determine the number of selected channels/kernels and often results in more compact models with better performance. Extensive experiments on both image classification and face recognition demonstrate the effectiveness of our methods. For example, on ILSVRC-12, the resultant ResNet-50 model with 30% reduction of channels even outperforms the baseline model by 0.36% in terms of Top-1 accuracy. The pruned MobileNetV1 and MobileNetV2 achieve 1.93× and 1.42× inference acceleration on a mobile device, respectively, with negligible performance degradation. The source code and the pre-trained models are available at https://github.com/SCUT-AILab/DCP.