  Let F_{q^n} be the field of order q^n, and let Tr be the trace map from
F_{q^n} to its q-element subfield. We exhibit nine sequences of polynomials of
the form f(x):=x+c*Tr(x^k), with c in F_{q^n}, such that for each polynomial
the function F_{q^n}-->F_{q^n} given by c-->f(c) is a permutation of F_{q^n}.
We also computed all permutation polynomials of this form over finite fields of
size less than 5000, and found that our examples comprise all examples with n>1
except for some simple cases where the polynomial induces a homomorphism of the
additive group of F_{q^n}, along with a few sporadic examples. One intriguing
feature is that our proofs of the different sequences use various different
methods, including a new variant of Dobbertin's method among others.
