  We present a brief survey of the recent theoretical work related to generic
Heisenberg spin models describing quasi-one-dimensional quantum ferrimagnets.
The emphasis is on quantum chains and ladders with strong competing
interactions, such as the frustrated $J_1-J_2$ chain with alternating (1,1/2)
spins, the spin-1/2 diamond chain with four-spin cyclic couplings, and some
generic types of mixed-spin ladders with geometric frustration. As a rule,
discussed models exhibit rich quantum phase diagrams and provide some
interesting examples of one-dimensional magnetic-paramagnetic quantum phase
transitions. A number of open problems in the reviewed research area are
discussed.
