  Product Retrieval (PR) and Grounding (PG), aiming to seek image and
object-level products respectively according to a textual query, have attracted
great interest recently for better shopping experience. Owing to the lack of
relevant datasets, we collect two large-scale benchmark datasets from Taobao
Mall and Live domains with about 474k and 101k image-query pairs for PR, and
manually annotate the object bounding boxes in each image for PG. As annotating
boxes is expensive and time-consuming, we attempt to transfer knowledge from
annotated domain to unannotated for PG to achieve un-supervised Domain
Adaptation (PG-DA). We propose a {\bf D}omain {\bf A}daptive Produc{\bf t}
S{\bf e}eker ({\bf DATE}) framework, regarding PR and PG as Product Seeking
problem at different levels, to assist the query {\bf date} the product.
Concretely, we first design a semantics-aggregated feature extractor for each
modality to obtain concentrated and comprehensive features for following
efficient retrieval and fine-grained grounding tasks. Then, we present two
cooperative seekers to simultaneously search the image for PR and localize the
product for PG. Besides, we devise a domain aligner for PG-DA to alleviate
uni-modal marginal and multi-modal conditional distribution shift between
source and target domains, and design a pseudo box generator to dynamically
select reliable instances and generate bounding boxes for further knowledge
transfer. Extensive experiments show that our DATE achieves satisfactory
performance in fully-supervised PR, PG and un-supervised PG-DA. Our
desensitized datasets will be publicly available
here\footnote{\url{https://github.com/Taobao-live/Product-Seeking}}.
