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This paper investigates representations of real numbers with an arbitrary negative base -- ββ < --1, which we call the (-- ββ)-expansions. They arise from the orbits of the (-- ββ)-transformation which is a natural modification of the ββ-transformation. We show some fundamental properties of (-- ββ)-expansions, each of which corresponds to a well-known fact of ordinary ββ-expansions. In particular, we characterize the admissible sequences of (-- ββ)-expansions, give a necessary and sufficient condition for the (-- ββ)- shift to be sofic, and explicitly determine the invariant measure of the (-- ββ)-transformations. [ABSTRACT FROM AUTHOR]