  We generalize the Bardeen-Cooper-Schrieffer-Bose-Einstein-condensation
(BCS-BEC) crossover of two-component fermions, which is realized by tuning the
$s$-wave scattering length $a$ between the fermions, to the case of an
arbitrary effective range $r_{\rm e}$. By using the Nozi\`{e}res-Schmitt-Rink
(NSR) approach, we show another crossover by changing $r_{\rm e}$ and present
several similarities and differences between these two crossovers. Furthermore,
the region ($r_{\rm e}>a/2$) where the effective range expansion breaks down
and the Hamiltonian becomes non-Hermitian is found, being consistent with the
Wigner's causality bound. Our results are universal for low-density interacting
fermions with low-energy constants $a$ and $r_{\rm e}$ and are directly
relevant to ultracold Fermi atomic gases as well as dilute neutron matter.
