What is the identity of Dark Matter? 
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*What is the identity of dark matter? 

*Is it a particle? 

*Is it the lightest superpartner (LSP)? 

*Do the phenomena attributed to dark matter point not to some form of matter but actually to an extension of gravity?
 A: At this point in time, your questions are subject to research - we just don't know, yet.
We see the effects of dark matter (in rotation curves of galaxies, in gravitational lensing and others), so there is "something". But what exactly that "something" constitutes is not clear.
There are several theories, and to think of dark matter as some kind of particle is the most favored theory at the moment. It's possible that it's some kind of supersymmetric particle, but till this time there is no evidence that supersymmetry is more than a mathematical construct. CERN and others are searching for supersymmetric particles, but the discovery of the Higgs at a rather high energy has constrained the possible supersymmetric models (to a degree that some scientists say supersymmetry or at least the "beautiful" flavours of supersymmetry are dead) - but as always in science, every theory that makes falsifiable predictions is a valid theory until there are enough experiments to make the assumption that it might be the correct answer, because it was not wrong in so many different cases...
To address your last question: All theories that postulate another model of gravity have a certain problem: general relativity. Most of these models have not provided a full applicable theory for the "whole universe" case and cosmology, it's often just a modification of newtons law of gravity. And GR is said to be one of the best tested theories (together with quantum theory), because it is so obscure everyone want's to prove Einstein wrong. But till today nobody could do that. Any alternative theory would have to stand against two criterions:
1. Make the same predictions for all these cases GRT and ART were successfully tested against
2. Make some predictions different from GRT (otherwise it would be the same theory using a different formalism)
The mentioned MOND theory is maybe the most advanced theory at that kind, but to my knowledge it still fails at certain points.
