Structures and properties of liquid crystalline phases formed by bent-core molecules are reviewed. At least eight phases designated as B1-B8 have been found, being unambiguously distinguished from phases formed by usual calamitic molecules due to a number of remarkable peculiarities. In addition to B1-B8 phases, smectic A-like phases and biaxial nematic phases formed by bent-core molecules are also reviewed. The most attractive aspects of this new class of liquid crystals are in polarity and chirality, despite being formed from achiral molecules. The bent-core mesogens are the first ferroelectric and antiferroelectric liquid crystals realized without introducing chirality. Spontaneous chiral deracemization at microscopic and macroscopic levels occurs and is controllable. Moreover, achiral bent-core molecules enhance system chirality. The interplay between polarity and chirality provides chiral nonlinear optic effects. Further interesting phenomena related to polarity and chirality are also reviewed.