  The one dimensional (1D) spin-1/2 Heisenberg-Ising model, a prototype quantum
many-body system, has been intensively studied for many years. In this review,
after a short introduction on some basic concepts of group theory for the
octahedral group, a detailed pedagogical framework is laid down to derive the
low-energy effective Hamiltonian for the Co-based materials. The 1D spin-1/2
Heisenberg-Ising model is obtained when applying the analysis to quasi-1D
antiferromagnetic materials $\rm BaCo_2V_2O_8$ and $\rm SrCo_2V_2O_8$. After
the preparation, we review the theoretical progresses of a variety of novel
magnetic excitations and emergent physics in the 1D spin-1/2 Heisenberg-Ising
model, and further summarize their recent experimental realizations.
