  We present the first systematic search for microlensing events with
variability in their baselines using data from the third phase of the Optical
Gravitational Lensing Experiment (OGLE-III). A total of 137 candidates (88 new)
was discovered toward the Galactic bulge. Among these, 21 have periodic
oscillations in their baselines, 111 are irregular variables and 5 are
potential long period detached eclipsing binaries. This is about 10% of the
total number of constant baseline events. They are hence quite common and can
be regarded as a new type of exotic events, which allow the determination of
extra parameters of the events. We show that microlensing of variable stars may
allow us to break the degeneracy between the blending parameter and
magnification. We note that in some cases variability hidden in the baseline
due to strong blending may be revealed in highly magnified events and resemble
other exotic microlensing behavior, including planetary deviation. A new system
(VEWS) for detecting ongoing variable baseline microlensing events is
presented.
