  We study first-order logic over unordered structures whose elements carry a
finite number of data values from an infinite domain. Data values can be
compared wrt.\ equality. As the satisfiability problem for this logic is
undecidable in general, we introduce a family of local fragments. They restrict
quantification to the neighbourhood of a given reference point that is bounded
by some radius. Our first main result establishes decidability of the
satisfiability problem for the local radius-1 fragment in presence of one
"diagonal relation". On the other hand, extending the radius leads to
undecidability. In a second part, we provide the precise decidability and
complexity landscape of the satisfiability problem for the existential
fragments of local logic, which are parameterized by the number of data values
carried by each element and the radius of the considered neighbourhoods.
Altogether, we draw a landscape of formalisms that are suitable for the
specification of systems with data and open up new avenues for future research.
