It is proposed to make this paper the first of a series of three dealing with the quantum physics of solids. This one will be concerned with the quantum states of electrons in crystals. The discussion will commence with an introductory section devoted to the failure of classical physics to account for phenomena of an atomic scale. Next, the quantum theory of electrons in atoms will be discussed, together with the resultant explanation of the structure of the periodic table; this is designed to illustrate the meaning of various quantum mechanical ideas which are important in understanding solids. Furthermore, much of the detailed information about atomic quantum states of particular atoms will be needed in the later discussion of the properties of certain solids. As an introduction to the modification of the quantum states occurring when atoms are put together to form a crystal, a short section will be devoted to structure of diatomic molecules. The next section will be concerned with quantum states for electrons in crystals. Whereas in an atom there are a series of isolated energies possible for an electron (corresponding to the various quantum states), in a crystal there are bands of allowed energies. This concept of energy bands is essential to the theory of crystals in much the same way that the concept of energy levels is essential to that of atoms. In terms of energy bands, the energy holding crystals together can be interpreted on a common basis for a wide variety of crystal types. This will be followed by a brief description of various crystal types and by a discussion of thermal properties in which the smallness of the electronic specific heat will be shown. The last section will be devoted to a discussion of para and ferromagnetism on the basis of the energy band picture. In the second paper, problems connected with electric currents and the motion of electrons through crystals will be discussed. This leads to the concept of the Brillouin zone which is complementary to that of the energy band, the two together forming the basis for discussing the quantum states of electrons in crystals. The third paper of the series will contain a comparison between theory and experiment for the alkali metals, the principal emphasis being placed upon the physical picture of the state of affairs in these simple metals.