Conventional medical image segmentation methods have been found inadequate in facilitating physicians with the identification of specific lesions for diagnosis and treatment. Given the utility of text as an instructional format, we introduce a novel task termed Medical Image Referring Segmentation (MIRS), which requires segmenting specified lesions in images based on the given language expressions. Due to the varying object scales in medical images, MIRS demands robust vision-language modeling and comprehensive multi-scale interaction for precise localization and segmentation under linguistic guidance. However, existing medical image segmentation methods fall short in meeting these demands, resulting in insufficient segmentation accuracy. In response, we propose an approach named Language-guided Scale-aware MedSegmentor (LSMS), incorporating two appealing designs: (1)~a Scale-aware Vision-Language Attention module that leverages diverse convolutional kernels to acquire rich visual knowledge and interact closely with linguistic features, thereby enhancing lesion localization capability; (2)~a Full-Scale Decoder that globally models multi-modal features across various scales, capturing complementary information between scales to accurately outline lesion boundaries. Addressing the lack of suitable datasets for MIRS, we constructed a vision-language medical dataset called Reference Hepatic Lesion Segmentation (RefHL-Seg). This dataset comprises 2,283 abdominal CT slices from 231 cases, with corresponding textual annotations and segmentation masks for various liver lesions in images. We validated the performance of LSMS for MIRS and conventional medical image segmentation tasks across various datasets. Our LSMS consistently outperforms on all datasets with lower computational costs. The code and datasets will be released.