Quantizing deep neural networks is an effective method for reducing memory consumption and improving inference speed, and is thus useful for implementation in resource-constrained devices. However, it is still hard for extremely low-bit models to achieve accuracy comparable with that of full-precision models. To address this issue, we propose learnable companding quantization (LCQ) as a novel non-uniform quantization method for 2-, 3-, and 4-bit models. LCQ jointly optimizes model weights and learnable companding functions that can flexibly and non-uniformly control the quantization levels of weights and activations. We also present a new weight normalization technique that allows more stable training for quantization. Experimental results show that LCQ outperforms conventional state-of-the-art methods and narrows the gap between quantized and full-precision models for image classification and object detection tasks. Notably, the 2-bit ResNet-50 model on ImageNet achieves top-1 accuracy of 75.1% and reduces the gap to 1.7%, allowing LCQ to further exploit the potential of non-uniform quantization.