Built on deep networks, end-to-end optimized image compression has made impressive progress in the past few years. Previous studies usually adopt a compressive auto-encoder, where the encoder part first converts image into latent features, and then quantizes the features before encoding them into bits. Both the conversion and the quantization incur information loss, resulting in a difficulty to optimally achieve arbitrary compression ratio. We propose iWave++ as a new end-to-end optimized image compression scheme, in which iWave, a trained wavelet-like transform, converts images into coefficients without any information loss. Then the coefficients are optionally quantized and encoded into bits. Different from the previous schemes, iWave++ is versatile: a single model supports both lossless and lossy compression, and also achieves arbitrary compression ratio by simply adjusting the quantization scale. iWave++ also features a carefully designed entropy coding engine to encode the coefficients progressively, and a de-quantization module for lossy compression. Experimental results show that lossy iWave++ achieves state-of-the-art compression efficiency compared with deep network-based methods; on the Kodak dataset, lossy iWave++ leads to 17.34 percent bits saving over BPG; lossless iWave++ achieves comparable or better performance than FLIF. Our code and models are available at https://github.com/mahaichuan/Versatile-Image-Compression.