Step 1: Recall the Biological Species Concept.
The BSC emphasizes \emph{reproductive isolation} as the defining criterion for species. It works well for many sexually reproducing organisms but has limitations in certain cases.
Step 2: Evaluate each option.
(A) \emph{Fertile interspecies hybrids.}
Yes — if two populations can interbreed and produce fertile hybrids (e.g., certain ducks, plants), the reproductive isolation criterion is blurred. This challenges the BSC.
(B) \emph{Extinct fossil species.}
Yes — the BSC relies on reproductive isolation, which cannot be tested for fossils. Thus, extinct species are usually defined using morphological (morphospecies) concepts, not the BSC.
(C) \emph{Barriers to gene flow.}
Not necessarily a challenge — barriers to gene flow (e.g., rivers, mountains) \emph{create} reproductive isolation, which supports the BSC rather than undermining it.
(D) \emph{Inbreeding depression.}
No — inbreeding depression reduces fitness within a population but does not challenge the definition of species boundaries under the BSC.
Step 3: Conclusion.
The main challenges to the BSC are cases where reproductive isolation cannot be applied or is ambiguous, such as fertile hybrids and fossil species.
Final Answer:\quad \(\boxed{\text{(A) and (B)}}\)