What about the radioactive waste? All nuclear reactors produce waste (finally when disassembled after the reactors' end of life). I find it strange to search for more efficient reactor styles, while the waste problem is not solved.
The idea of the liquid fueled reactors is that you can easily change the fuel composition, ideally constantly reprocessing the fuel of contaminates, while letting the unburned fuel stay in place. This would drastically reduce the amount of waste generated (by orders of magnitude), and the 'unburnable' waste left over actually has some uses of its own (molybdenum 99 and bismuth 213).
What hasn't been solved is the effects of of all our coal, gas, and oil usage that could be replaced by nuclear power. It makes all the sense in the world to be finding ways we can widely and safely deploy nuclear power worldwide to replace coal and gas power plants. Generating clean electricity is a requisite for wide spread electric or hydrogen cars to stop burning gasoline.
If you believe that coal and gas burning is bad for the environment, then we need nuclear power urgently and widely, energy needs are only going to go up and the only alternative to nuclear is burning more fossil fuels. I find it absolutely absurd that people worry more about nuclear waste than atmosphere pollution currently happening. I think we could store a few hundred years of nuclear waste on Antarctica til technology figures out what to do with it, even if that's just shooting it into the sun. I choose safely storing nuclear waste for the future to deal with over continuing to burn billions of tons of coal and gas each year which is actively killing about a million people per year. (WHO 2008)
Thus, solving the problem once and for all. ONCE AND FOR ALL!
The waste in conventional reactors are of two types, fission procucts and higher actinides. The fision products are the result of splitted uranium atoms. These are intensively radioactive, with halflives from milliseconds to about 30 years.
Then there are higher actinides, with halflives up to tens of thousands of years. These are less radioactive, but still radioactive enough that they have to be safely stored. This storage must be safe for hundred of thousands of years, something that it is hard to guarantee. Fision products only need to be stored for about 600-800 years before they are no more radioactive than uranium ore. Thorium rectors produce virtually no higher actinides, so it is easier to find storage that is safe for the period it must stay out of reach. It is also less volume of the waste (but more concentrated, as the amount of radioactivity is about the same), since the fission products are not mixed with U238 as in conventional reactors. This means that less storage is needed, which also makes it easier to handle.
And the currently used alternatives (coal) produces way more waste (whole mountains of slightly radioactive, and slightly toxic dust leftovers from coal extraction build up near coal mines).