Quantizing weights and activations of deep neural networks is essential for deploying them in resourceconstrained devices, or cloud platforms for at-scale services. While binarization is a special case of quantization, this extreme case often leads to several training difficulties, and necessitates specialized models and training methods. As a result, recent quantization methods do not provide binarization, thus losing the most resource-efficient option, and quantized and binarized networks have been distinct research areas. We examine binarization difficulties in a quantization framework and find that all we need to enable the binary training are a symmetric quantizer, good initialization, and careful hyperparameter selection. These techniques also lead to substantial improvements in multibit quantization. We demonstrate our unified quantization framework, denoted as UniQ, on the ImageNet dataset with various architectures such as ResNet-18,-34 and Mo-bileNetV2. For multi-bit quantization, UniQ outperforms existing methods to achieve the state-of-the-art accuracy. In binarization, the achieved accuracy is comparable to existing state-of-the-art methods even without modifying the original architectures.