Modern speaker verification (SV) systems typically demand expensive storage and computing resources, thereby hindering their deployment on mobile devices. In this paper, we explore adaptive neural network quantization for lightweight speaker verification. Firstly, we propose a novel adaptive uniform precision quantization method which enables the dynamic generation of quantization centroids customized for each network layer based on k-means clustering. By applying it to the pre-trained SV systems, we obtain a series of quantized variants with different bit widths. To enhance the performance of low-bit quantized models, a mixed precision quantization algorithm along with a multi-stage fine-tuning (MSFT) strategy is further introduced. Unlike uniform precision quantization, mixed precision approach allows for the assignment of varying bit widths to different network layers. When bit combination is determined, MSFT is employed to progressively quantize and fine-tune network in a specific order. Finally, we design two distinct binary quantization schemes to mitigate performance degradation of 1-bit quantized models: the static and adaptive quantizers. Experiments on VoxCeleb demonstrate that lossless 4-bit uniform precision quantization is achieved on both ResNets and DF-ResNets, yielding a promising compression ratio of around 8. Moreover, compared to uniform precision approach, mixed precision quantization not only obtains additional performance improvements with a similar model size but also offers the flexibility to generate bit combination for any desirable model size. In addition, our suggested 1-bit quantization schemes remarkably boost the performance of binarized models. Finally, a thorough comparison with existing lightweight SV systems reveals that our proposed models outperform all previous methods by a large margin across various model size ranges.