This article proposes a voice conversion (VC) method using sequence-to-sequence (seq2seq or S2S) learning, which flexibly converts not only the voice characteristics but also the pitch contour and duration of input speech. The proposed method, called ConvS2S-VC, has three key features. First, it uses a model with a fully convolutional architecture. This is particularly advantageous in that it is suitable for parallel computations using GPUs. It is also beneficial since it enables effective normalization techniques such as batch normalization to be used for all the hidden layers in the networks. Second, it achieves many-to-many conversion by simultaneously learning mappings among multiple speakers using only a single model instead of separately learning mappings between each speaker pair using a different model. This enables the model to fully utilize available training data collected from multiple speakers by capturing common latent features that can be shared across different speakers. Owing to this structure, our model works reasonably well even without source speaker information, thus making it able to handle any-to-many conversion tasks. Third, we introduce a mechanism, called the conditional batch normalization that switches batch normalization layers in accordance with the target speaker. This particular mechanism has been found to be extremely effective for our many-to-many conversion model. We conducted speaker identity conversion experiments and found that ConvS2S-VC obtained higher sound quality and speaker similarity than baseline methods. We also found from audio examples that it could perform well in various tasks including emotional expression conversion, electrolaryngeal speech enhancement, and English accent conversion.