The contemporary theoretical ideas on quantum diffusion of impurity particles and point defects in crystals of solid helium are reviewed. Results are given of experimental studies on diffusion of isotopic impurities and ions in solid helium in which quantum diffusion was actually found. The energy spectrum of a quantum crystal in the presence of impurities or point defects is discussed. Owing to the specific dynamics of impurity particles, excitations of the crystal exist that behave like one- or two-dimensional quasiparticles within the volume of the three-dimensional crystal. Experimental nuclear magnetic resonance data in solid solutions of helium isotopes confirm the existence of two-dimensional quasiparticles. The features of behavior of dislocations in quantum crystals are elucidated. Experiments are discussed in which these features should be manifested.