Let us clearly draw the line between two things here, since the question can easily involve opinion based answers, which may also be dubbed non-mainstream (which isn't welcome on this site).
1) The existence of dark matter is generally believed by a majority of the Physics community, since astronomical observations, notably by the Planck space observatory, suggest that visible matter makes up only $\sim 5 \%$ of the total mass energy of the universe. Most conveniently, that means that some matter isn't visible to us, hence we call it 'dark', and look at extensions of current theoretical frameworks to account for them, as <PERSON> and <PERSON> have mentioned in their answers.
However, there are alternatives, which are intended towards accounting for these observations without including any extra invisible matter, such as the Modified Newtonian Dynamics Approach. If this is proved correct with time, we won't need to worry about the dark-matter particles you are asking about. The final answer is really far off, since neither of these mentioned approaches can claim to be the final word on this subject.
2) Regarding ''Dark Matter Particles'', as <PERSON> mentions in a comment, one can't be absolutely sure as to how to look for them with the current set of theoretical understanding, yet I am aware of certain experimental groups having reported measurements relating to WIMPs, such as the widely popularized Super Cryogenic Dark Matter Search (SuperCDMS) experiment. The strategy here is to measure the recoil energy of a nucleus in a Nucleus-WIMP collision, which is more or less a direct detection as compared to the indirect method - analyzing the debris in WIMP decays or annihilations. 70
They had reported some measurements, at roughly $3\sigma$ level, which are encouraging signs, though not conclusive yet. (An alternative reference can be found here ). (Sidenote - The trust level is a little higher than $3\sigma$; nobody believed the Higgs boson ''discovery'', till a $5\sigma$ measurement was reported.)
A completely consistent theory of their genesis, is an issue that can be settled only after we know for sure that they exist. Right now, neither the theoretical nor the experimental status of this problem can warrant that. Or worst case, if MoND works out fine, we may not even need them. Only time will tell. 1020