Text-guided style transfer aims to repaint a content image with the target style described by a text prompt, offering greater flexibility and creativity compared to traditional image-guided style transfer. Despite the potential, existing text-guided style transfer methods often suffer from many issues, including insufficient visual quality, poor generalization ability, or a reliance on large amounts of paired training data. To address these limitations, we leverage the inherent strengths of transformers in handling multimodal data and propose a novel transformer-based framework called TRTST that not only achieves unpaired arbitrary text-guided style transfer but also significantly improves the visual quality. Specifically, TRTST explores combining a text transformer encoder with an image transformer encoder to project the input text prompt and content image into a joint embedding space and extract the desired style and content features. These features are then input into a multimodal co-attention module to stylize the image sequence based on the text sequence. We also propose a new adaptive parametric positional encoding (APPE) scheme which can adaptively produce different positional encodings to optimally match different inputs with a position encoder. In addition, to further improve content preservation, we introduce a text-guided identity loss to our model. Extensive results and comparisons are conducted to demonstrate the effectiveness and superiority of our method.