  The growing interest of different entropy functions proposed so far (like the
Bekenstein-Hawking, Tsallis, R\'{e}nyi, Barrow, Sharma-Mittal, Kaniadakis and
Loop Quantum Gravity entropies) towards black hole thermodynamics as well as
towards cosmology lead to the natural question that whether there exists a
generalized entropy function that can generalize all these known entropies.
With this spirit, we propose a new 4-parameter entropy function that seems to
converge to the aforementioned known entropies for certain limits of the
entropic parameters. The proposal of generalized entropy is extended to
non-singular case, in which case, the entropy proves to be singular-free during
the entire cosmological evolution of the universe. The hallmark of such
generalized entropies is that it helps us to fundamentally understand one of
the important physical quantities namely ``entropy''. Consequently we address
the implications of the generalized entropies on black hole thermodynamics as
well as on cosmology, and discuss various constraints of the entropic
parameters from different perspectives.
