While recent progress in video-text retrieval has been advanced by the exploration of better representation learning, in this paper, we present a novel multi-space multi-grained supervised learning framework, SUMA, to learn an aligned representation space shared between the video and the text for video-text retrieval. The shared aligned space is initialized with a finite number of concept clusters, each of which refers to a number of basic concepts (words). With the text data at hand, we are able to update the shared aligned space in a supervised manner using the proposed similarity and alignment losses. Moreover, to enable multi-grained alignment, we incorporate frame representations for better modeling the video modality and calculating fine-grained and coarse-grained similarity. Benefiting from learned shared aligned space and multi-grained similarity, extensive experiments on several video-text retrieval benchmarks demonstrate the superiority of SUMA over existing methods.