  Properties of the zigzag spin chains with various nearest-neighbor and
next-nearest-neighbor interactions are studied by making use of the
transfer-matrix renormalization group method. Thermodynamic quantities of the
systems (temperature dependence of the susceptibility and the specific heat),
as well as the field dependence of the magnetization are analyzed numerically
with a high accuracy in the thermodynamic limit. The results have been compared
with the recent experimental data on Rb$_2$Cu$_{2}$Mo$_{3}$O$_{12}$.
