The advantage is mainly that more active fuel can be created than is used, so getting 'something for nothing'. However experimental reactors such as the sodium cooled fast reactor at Dounreay in Scotland have proved very difficult in practice, with material problems, and as far as I know such experiments are in abeyance at the moment.
No, a breeder nuclear reactor does not typically use a moderator. Breeder reactors are designed to produce more fissile material than they consume by using fast neutrons to convert non-fissile isotopes into fissile ones without slowing down the neutrons.
Moderators are not used in a breeder reactor because their primary purpose is to slow down neutrons to increase the likelihood of fission events in a thermal reactor. In a breeder reactor, fast neutrons are required to convert non-fissile uranium-238 into fissile plutonium-239, so using a moderator would hinder this process.
The idea is to use it for nuclear fuel in other reactors, this is why it is called a breeder process. Uranium-238 which is not fissile is converted to Pu-239 in the breeder reactor, but of course there is a chemical separation process to go through before the Pu is available.
The Kalpakkam nuclear power plant uses heavy water (deuterium oxide) as a coolant and moderator in its Pressurized Heavy Water Reactors (PHWRs). Heavy water has a higher density than regular water, allowing it to efficiently slow down neutrons and regulate the nuclear reaction in the reactor core.
The KAMINI (Kalpakkam Mini) reactor is a Uranium-233 fueled, demineralized light water moderated and cooled, beryllium oxide reflected, low power nuclear research reactor. It is located in the post irradiation examination facility of Radio Metallurgy Laboratory, Indira Gandhi Centre for Atomic Research, Kalpakkam, India. A link is provided to the Wikipedia article.
Dounreay PFR (Prototype Fast Reactor)
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Kalpakkam has both Thermal reactor (Madras Atomic Power Station) and Fast Breeder Reactor -FBTR and PFBR (Under construction). In MAPS (thermal reactor) it is Heavy Water (D2O) Which acts as a coolant as well as moderator, where as in Fast Breeder Test Reactor (FBTR) and Prototype Fast Breeder Reactor (PFBR) Sodium (Na) is used as coolant. Since there should not any moderators for fast reactors D2O will not be used as coolant in fast reactors.
You may mean FBR - this stands for Fast Breeder Reactor
This is at Kalpakkam, I don't know which state. See Link below and go to section on Thorium Fuel Cycle
See the India section of www.world-nuclear.org
Moderator is not used in case of fast breeder reactor because there is no need to slow down neutron energy. Nuclear fission takes place at high energy of neutrons.
GEAR
The main disadvantage is the positive temperature coefficient of reactivity.