Learning-based point cloud (PC) compression is a promising research avenue to reduce the transmission and storage costs for PC applications. Existing learning-based methods to compress PCs have mainly focused on geometry and employ variational autoencoders to learn compact signal representations. However, autoencoders leverage low-dimensional bottlenecks that limit the maximum reconstruction quality, even at high bitrates. In this paper, we propose a different and novel approach to compress PC attributes by using normalizing flows. Since normalizing flows model invertible transforms, the proposed approach can achieve better reconstruction quality than variational autoencoders over a large range of bitrates. Our Normalizing Flow-based Point Cloud Attribute Compression (NF-PCAC) outperforms previous learning-based methods for attribute compression, and has comparable performance as G-PCC v.14, showing the potential of this scheme for PC compression.