Flow-based methods have demonstrated promising results in addressing the ill-posed nature of super-resolution (SR) by learning the distribution of high-resolution (HR) images with the normalizing flow. However, these methods can only perform a predefined fixed-scale SR, limiting their potential in real-world applications. Meanwhile, arbitrary-scale SR has gained more attention and achieved great progress. Nonetheless, previous arbitrary-scale SR methods ignore the ill-posed problem and train the model with per-pixel L1 loss, leading to blurry SR outputs. In this work, we propose "Local Implicit Normalizing Flow" (LINF) as a unified solution to the above problems. LINF models the distribution of texture details under different scaling factors with normalizing flow. Thus, LINF can generate photo-realistic HR images with rich texture details in arbitrary scale factors. We evaluate LINF with extensive experiments and show that LINF achieves the state-of-the-art perceptual quality compared with prior arbitrary-scale SR methods.