Abstract:  In this lecture the following problem is investigated: Let R be
a ring (mainly the ring of integers in a number field or function
field). When is it possible to represent all r ∈ R as a sum of
units: r= u_{1}+... + u_{k}, u_{i}∈ R^{*}. We
give sufficient conditions for numbers fields of low order and
present a general negative result of Jarden and Narkiewicz. There
are also many open problems which are discussed. The proofs depend on
tools from algebraic number theory and on diophantine analysis, in
particular, on the solution of Sunit equations. Furthermore we
establish asymptotic formulas for the number of such representations.
