Theoretical investigations of solitons arising in quasi-one-dimensional molecular structures such as alphahelical protein molecules are described. The properties of solitons and excitons are discussed. The reasons for the great stability of solitons are investigated. It is noted that the vibrational energy of Amide I vibrations of peptide groups can be transported in the form of solitons along a protein molecule without losses due to creation of phonons. Deceleration of solitons by dissipative forces and the effect of thermal motion and external fields are investigated. A new mechanism for muscle contraction in animals, based on the idea of solitons, is discussed on a molecular level.