arXiv:0811.4684 [ps, pdf, other] Title: Six Puzzles for LCDM Cosmology Authors: L. Perivolaropoulos Comments: 9 pages, 3 figures. Invited article to the TSPU anniversary volume "The Problems of Modern Cosmology" on the occasion of the 50th birthday of Prof. S. D. Odintsov Subjects: Astrophysics (astro-ph) Quote. The LCDM cosmological model is a well defined, simple and predictive model which is consistent with the majority of current cosmological observations. Despite of these successes there are specific cosmological observations which differ from the predictions of LCDM at a level of 2\sigma or higher. These observations include the following: 1. Large Scale Velocity Flows (LCDM predicts significantly smaller amplitude and scale of flows than what observations indicate), 2. Brightness of Type Ia Supernovae (SnIa) at High Redshift z (LCDM predicts fainter SnIa at High z), 3. Emptiness of Voids (LCDM predicts more dwarf or irregular galaxies in voids than observed), 4. Profiles of Cluster Haloes (LCDM predicts shallow low concentration and density profiles in contrast to observations which indicate denser high concentration cluster haloes) 5. Profiles of Galaxy Haloes (LCDM predicts halo mass profiles with cuspy cores and low outer density while lensing and dynamical observations indicate a central core of constant density and a flattish high dark mass density outer profile), 6. Sizable Population of Disk Galaxies (LCDM predicts a smaller fraction of disk galaxies due to recent mergers expected to disrupt cold rotationally supported disks). Even though the origin of some of the above challenges may be astrophysical or related to dark matter properties, it should be stressed that even on galactic and cluster scales, the effects of dark energy on the equilibrium and stability of astrophysical systems are not negligible and they may play a key role in the resolution of the above puzzles. Here, I briefly review these six challenges of LCDM and discuss the possible dark energy properties required for their resolution.