Arbitrary-shaped scene text detection is still a challenging task in computer vision, due to its high versatility and complexity. Most methods directly describe text instances by rectangles, quadrangles, or curves, probably leading to insufficient representations. We introduce wavelet descriptor (WD) for arbitrary-shaped text representation that decomposes text contours to wavelet coefficients and extracts feature sequences by discrete wavelet transform (DWT). Based on WD, we propose WD network, which predicts wavelet coefficients then reconstructs text instances accurately and smoothly via inverse DWT and nonmaximum suppression. We further study the restoration capability of various wavelet functions for text contour, including sym5, bior3.1, bior4.4, coif3, db5, dmey, and rbio2.8. To evaluate the proposed method, we conduct experiments on available public datasets. The F-measure of our method achieves 85.8% on challenging curved dataset CTW1500 and higher than 85% on Total-Text and ICDAR2015, demonstrating the excellent performance of our method.