  I have carried out a project to study the eccentric neutron-star binary
Circinus X-1 through an extensive series of observational studies with the
Rossi X-ray Timing Explorer satellite and through theoretical computer models I
developed to explore mass transfer and evolution in an eccentric binary. I
modeled the evolution of the energy spectrum during intensity dips with a
variably absorbed bright component plus a fainter unabsorbed component. I show
that variability not attributable to absorption dips is related to the
spectral/intensity states of the ``Z source'' class of low-mass X-ray binaries
(LMXBs), namely motion along (or shifts of) the horizontal, normal, and flaring
branches of the ``Z'' track in color-color and hardness-intensity diagrams. I
found rapid X-ray variability properties associated with each
spectral/intensity state: On the horizontal branch, quasi-periodic oscillations
(QPOs) in the X-ray intensity shift in frequency from 1.3 to 35 Hz. On the
normal branch, a different QPO occurs at about 4 Hz. On the flaring branch only
strong aperiodic variability occurs. I also modeled the evolution of the energy
spectra associated with each of these branches. To study mass transfer in an
eccentric binary, I developed computer codes for transfer via Roche-lobe
overflow and from a stellar wind. I derive theoretical mass accretion profiles
and compare them to the observed profile of the X-ray intensity. In order to
explore the possible evolutionary history of Circinus X-1, I developed a
binary-evolution computer code for a neutron-star and low-mass companion in an
eccentric orbit. I use this code in a population-synthesis study to show that
the number of systems in the Galaxy expected to resemble Cir X-1 is of order
unity, consistent with its unique status as an LMXB with high eccentricity.
