Multi-bit spiking neural networks (SNNs) have recently become a heated research spot, pursuing energy-efficient and high-accurate AI. However, with more bits involved, the associated memory and computation demands escalate to the point where the performance improvements become disproportionate. Based on the insight that different layers demonstrate different importance and extra bits could be wasted and interfering, this paper presents an adaptive bit allocation strategy for direct-trained SNNs, achieving fine-grained layer-wise allocation of memory and computation resources. Thus, SNN's efficiency and accuracy can be improved. Specifically, we parametrize the temporal lengths and the bit widths of weights and spikes, and make them learnable and controllable through gradients. To address the challenges caused by changeable bit widths and temporal lengths, we propose the refined spiking neuron, which can handle different temporal lengths, enable the derivation of gradients for temporal lengths, and suit spike quantization better. In addition, we theoretically formulate the step-size mismatch problem of learnable bit widths, which may incur severe quantization errors to SNN, and accordingly propose the step-size renewal mechanism to alleviate this issue. Experiments on various datasets, including the static CIFAR and ImageNet datasets and the dynamic CIFAR-DVS and DVS-GESTURE datasets, demonstrate that our methods can reduce the overall memory and computation cost while achieving higher accuracy. Particularly, our SEWResNet-34 can achieve a 2.69\% accuracy gain and 4.16$\times$ lower bit budgets over the advanced baseline work on ImageNet. This work is open-sourced at \href{https://github.com/Ikarosy/Towards-Efficient-and-Accurate-Spiking-Neural-Networks-via-Adaptive-Bit-Allocation}{this link}.