  This is a largely expository article based on our previous work on arithmetic
diagonal cycles on unitary Shimura varieties. We define a class of Shimura
varieties closely related to unitary groups which represent a moduli problem of
abelian varieties with additional structure, and which admit interesting
algebraic cycles. We generalize to arbitrary signature type the results of our
previous work valid under special signature conditions. We compare our Shimura
varieties with other unitary Shimura varieties.
