Our calculations show that a nonchiral nanocrystal is able to dramatically change the circular dichroism (CD) of a chiral molecule when the nanocrystal and molecule form a complex and couple via dipole and multipole Coulomb interactions. Plasmon resonances of metal nanocrystals in the nanocrystal-molecule complex result in both the resonant enhancement of CD signals of molecules and the appearance of new spectral structures. Two mechanisms, in which a nanocrystal can influence the CD effect, have been identified. The first mechanism is the plasmon-induced change in the electromagnetic field inside the chiral molecule. The second is the optical absorption of the nanocrystal-molecule complex due to the chiral currents inside the metal nanocrystal induced by the dipole of the chiral molecule. The first mechanism creates a change in the angle between the effective electric and magnetic dipoles of the molecule. This mechanism can lead to symmetry breaking and to a plasmon-induced CD signal of the nonchiral molecule. Both mechanisms create interesting Fano-like shapes in the CD spectra. Importantly, the second mechanism gives the main contribution to the CD signal at the plasmon frequency when the absorption band of the chiral molecule is far from the plasmon resonance. This may happen in many cases since many biomolecules are optically active in the UV range, whereas plasmon resonances in commonly used nanometals are found at longer wavelengths. As concrete examples, the paper describes alpha-helix and calixarene ligand molecules coupled with metal nanocrystals. The above results are also applied to complexes incorporating semiconductor nanocrystals. The results obtained here can be used to design a variety of hybrid nanostructures with enhanced and tailored optical chirality in the visible wavelength range.