Audio captioning aims to automatically generate a natural language description of an audio clip. Most captioning models follow an encoder-decoder architecture, where the decoder predicts words based on the audio features extracted by the encoder. Convolutional neural networks (CNNs) and recurrent neural networks (RNNs) are often used as the audio encoder. However, CNNs can be limited in modelling temporal relationships among the time frames in an audio signal, while RNNs can be limited in modelling the long-range dependencies among the time frames. In this paper, we propose an Audio Captioning Transformer (ACT), which is a full Transformer network based on an encoder-decoder architecture and is totally convolution-free. The proposed method has a better ability to model the global information within an audio signal as well as capture temporal relationships between audio events. We evaluate our model on AudioCaps, which is the largest audio captioning dataset publicly available. Our model shows competitive performance compared to other state-of-the-art approaches.