  This review reports some key results in theoretical investigations on
configurations of lipid membranes and presents several challenges in this field
which involve (i) exact solutions to the shape equation of lipid vesicles; (ii)
exact solutions to the governing equations of open lipid membranes; (iii) neck
condition of two-phase vesicles in the budding state; (iv) nonlocal theory of
membrane elasticity; (v) relationship between symmetry and the magnitude of
free energy.
