  We report on the discovery of thermal and non-thermal X-rays from the shells
of the super bubble (SB) 30 Dor C in the Large Magellanic Cloud (LMC). The
X-ray morphology is a nearly circular shell with a radius of about 40 pc, which
is bright on the northern and western sides. The spectra of the shells are
different from region to region. The southern shell shows clear emission lines,
and is well fitted with a model of a thin-thermal plasma (kT = 0.21keV) in
non-equilibrium ionization (NEI) plus a power-law component. This thermal
plasma is located inside of the H alpha emission, which is the outer edge of
the shell of the SB. The northern and western sides of the SB are dim in H
alpha emission, but are bright in non-thermal (power-law) X-rays with a photon
index of 2.1-2.9. The non-thermal X-ray shell traces the outer boundary of the
radio shell. These features of thin-thermal and non-thermal X-rays are similar
to those of SN 1006, a prototype of synchrotron X-ray shell, but the
non-thermal component of 30 Dor C is about ten-times brighter than that of SN
1006. 30 Dor C is the first candidate of an extragalactic SB, in which
energetic electrons are accelerating in the shell. The age is much older than
that of SN 1006, and hence the particle acceleration time in this SB may be
longer than those in normal shell-like SNRs. We found point-like sources
associated with some of tight star clusters. The X-ray luminosity and spectrum
are consistent with those of young clusters of massive stars. Point-like
sources with non-thermal spectra are also found in the SB. These may be
background objects (AGNs) or stellar remnants (neutron stars or black holes).
