  We intend to provide a comprehensive answer to the question on whether all
Coronal Mass Ejections (CMEs) have flux rope structure. To achieve this, we
present a synthesis of the LASCO CME observations over the last sixteen years,
assisted by 3D MHD simulations of the breakout model, EUV and coronagraphic
observations from STEREO and SDO, and statistics from a revised LASCO CME
database. We argue that the bright loop often seen as the CME leading edge is
the result of pileup at the boundary of the erupting flux rope irrespective of
whether a cavity or, more generally, a 3-part CME can be identified. Based on
our previous work on white light shock detection and supported by the MHD
simulations, we identify a new type of morphology, the `two-front' morphology.
It consists of a faint front followed by diffuse emission and the bright
loop-like CME leading edge. We show that the faint front is caused by density
compression at a wave (or possibly shock) front driven by the CME. We also
present high-detailed multi-wavelength EUV observations that clarify the
relative positioning of the prominence at the bottom of a coronal cavity with
clear flux rope structure. Finally, we visually check the full LASCO CME
database for flux rope structures. In the process, we classify the events into
two clear flux rope classes (`3-part', `Loop'), jets and outflows (no clear
structure). We find that at least 40% of the observed CMEs have clear flux rope
structures. We propose a new definition for flux rope CMEs (FR-CMEs) as a
coherent magnetic, twist-carrying coronal structure with angular width of at
least 40 deg and able to reach beyond 10 Rsun which erupts on a time scale of a
few minutes to several hours. We conclude that flux ropes are a common
occurrence in CMEs and pose a challenge for future studies to identify CMEs
that are clearly not FR-CMEs.
