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Abridged: Getting a better understanding of the evolution and nucleosyntheticyields of the most metal-poor stars (appr. Z<=10^-5) is critical because theyare part of the big picture of the history of the primitive Universe. Yet manyof the remaining unknowns of stellar evolution lie in the birth, life, anddeath 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 theirfinal fates. The depth and efficiency of mixing episodes are critical todetermine the mass limits for the formation of electron-capture supernovae, butour knowledge of these phenomena is not complete because they are stronglyaffected by the choice of input physics. We also consider the alternativeSNI1/2 channel to form SNe out of the most metal-poor IMS. In this case, it iscritical to understand the thermally-pulsing AGB evolution until the latestages. Efficient second dredge-up and, later, third dredge-up episodes couldbe able to pollute stellar envelopes enough for the stars to undergo thermalpulses in a way very similar to that of higher initial Z objects. Inefficient2nd and/or 3rd dredge-up may leave an almost pristine envelope, unable tosustain strong stellar winds. This may allow the H-exhausted core to grow toM_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 forthe formation of SNe, we discuss existing nucleosynthetic yields of stars ofmetallicity Z<=10^-5, and present an example of nucleosynthetic calculationsfor a thermally-pulsing Super-AGB star of Z=10^-5. We compare theoreticalpredictions with observations of the lowest [Fe/H] objects detected. The reviewcloses by discussing current open questions as well as possible fruitfulavenues for future research.

Fates of the oldest intermediate-mass stars - Primordial to Extremely Metal-Poor AGB and Super-AGB Stars: White Dwarf or Supernova progenitors