  We summarize our theoretical findings on the ground-state phase diagram of
the spin-1/2 XXZ chain having competing nearest-neighbor (J1) and
antiferromagnetic next-nearest-neighbor (J2) couplings. Our study is mainly
concerned with the case of ferromagnetic J1, and the case of antiferromagnetic
J1 is briefly reviewed for comparison. The phase diagram contains a rich
variety of phases in the plane of J1/J2 versus the XXZ anisotropy Delta:
vector-chiral phases, N\'eel phases, several dimer phases, and
Tomonaga-Luttinger liquid phases. We discuss the vector-chiral order that
appears for a remarkably wide parameter space, successive N\'eel-dimer phase
transitions, and an emergent nonlocal string order in a narrow region of
ferromagnetic J1 side.
