  It has been known since the 1970's that the difference of the non-zero
weights of a projective $\mathbb{F}_q$-linear two-weight has to be a power of
the characteristic of the underlying field. Here we study non-projective
two-weight codes and e.g.\ show the same result under mild extra conditions.
For small dimensions we give exhaustive enumerations of the feasible parameters
in the binary case.
