  Abridged: Getting a better understanding of the evolution and nucleosynthetic
yields of the most metal-poor stars (appr. Z<=10^-5) is critical because they
are part of the big picture of the history of the primitive Universe. Yet many
of the remaining unknowns of stellar evolution lie in the birth, life, and
death of these objects. We review stellar evolution of intermediate-mass (IMS)
Z<=10-5 models existing in the literature, with a focus on the problem of their
final fates. The depth and efficiency of mixing episodes are critical to
determine the mass limits for the formation of electron-capture supernovae, but
our knowledge of these phenomena is not complete because they are strongly
affected by the choice of input physics. We also consider the alternative
SNI1/2 channel to form SNe out of the most metal-poor IMS. In this case, it is
critical to understand the thermally-pulsing AGB evolution until the late
stages. Efficient second dredge-up and, later, third dredge-up episodes could
be able to pollute stellar envelopes enough for the stars to undergo thermal
pulses in a way very similar to that of higher initial Z objects. Inefficient
2nd and/or 3rd dredge-up may leave an almost pristine envelope, unable to
sustain strong stellar winds. This may allow the H-exhausted core to grow to
M_Ch before the envelope is lost, and thus let the star explode as a SNI1/2.
After reviewing the information available on these two possible channels for
the formation of SNe, we discuss existing nucleosynthetic yields of stars of
metallicity Z<=10^-5, and present an example of nucleosynthetic calculations
for a thermally-pulsing Super-AGB star of Z=10^-5. We compare theoretical
predictions with observations of the lowest [Fe/H] objects detected. The review
closes by discussing current open questions as well as possible fruitful
avenues for future research.
