We deal with the basic convexity properties -rotundity, and uniform, l
ocal uniform and full rotundity - for symmetric spaces. A characteriza
tion of Orlicz-Lorentz spaces with the Kadec-Klee property for pointwi
se convergence is given. These results are applied to obtain criteria
of convexity properties for Orlicz-Lorentz sequence spaces, and some n
ew proofs of the sufficiency part of criteria for rotundity and unifor
m rotundity for Orlicz-Lorentz function spaces.