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.
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