There are two nuclear breeder reaction sequences
for which reactors have been built.

In these lines, +1n represents capture of a neutron, mass 1.
-> beta represents decay by emitting a beta particle, which is a high energy electron.
  1.    232Th+1n -> beta+ 233Pa -> beta+ 233U
  2.    238U +1n -> beta+ 239Np -> beta+ 239Pu

The snag with (1) is that in a single stage reactor, the protactinium 233Pa, with a half life of about 27 days, is exposed to neutrons that it can wastefully capture and become the rather useless 234Pa.
By comparison, 2.4 days is the half life for 239Np.
A more expensive two stage reactor, which protects the 233Pa from neutrons, is theoretically a possible source of weapons grade fissile uranium.

The snag in (2) is the widespread ignorant fear of plutonium, which was first used quite wantonly to destroy a city in Japan, three days after the first was destroyed with the only large concentration of 235U in existence.
Plutonium is not found in nature, and is apparently feared far more than radium, which occurs naturally. I presume this is the superstitious fear of things that "aren't natural".
Radium is not only 15 times as radioactive as plutonium, but its immediate descendant radon is thousands of times worse, and as a gas can easily enter the lungs, where its radioactive descendants will stay and wreak havoc.
Reactor grade plutonium is very nearly useless for bomb making, because its neutron capture isotope 240 is spontaneously fissile, dangerous both to the bomb makers, and in the prospect of pre-ignition of the bomb.

But note that in any case, a reactor that breeds fissile material is designed continually to consume the products it is manufacturing.
There is a distinctio made in some discussions, of burner or sustainer or breeder. The last of these would be a reactor designed to create startup fuel for more reactors.