In the early 20th century, Danish physicist Niels Bohr made many contributions to atomic theory and quantum physics. Among these are his model of the atom, which was an improved version of the previous atomic model by Ernest Rutherford. This is officially known as the Rutherford-Bohr model, but is often called the Bohr model for short.

Rutherford's model contained a compact, positively charged nucleus surrounded by a diffuse cloud of electrons. This naturally led to a planetary model of the atom, with the nucleus acting as the sun and the electrons as planets in circular orbits like a miniature solar system.

A key failure of this model, however, was that the electrons (unlike planets) had nonzero electric charge and would therefore radiate energy as they orbited the nucleus. This would lead to them falling into the center, radiating a "smear" of energies across the electromagnetic spectrum as they fell. But it was known that electrons had stable orbits, and their radiated energies occurred in discrete amounts called spectral lines.

Bohr's model was an extension of the Rutherford model and contained three postulates:

1. Electrons are able to move in certain discrete stable orbits without radiating energy.
2. These special orbits have angular momentum values that are integer multiples of the reduced Planck's constant ħ (sometimes called h-bar).
3. The electrons can only gain or lose very specific amounts of energy by jumping from one orbit to another in discrete steps, absorbing or emitting radiation of a specific frequency.

Bohr's model provides a good first-order approximation of energy levels for simple atoms such as the hydrogen atom.

An electron's angular momentum must be

where ​m​ is the mass of the electron, ​v​ is its velocity, ​r​ is the radius at which it orbits the nucleus and quantum number ​n​ is a non-zero integer. Since the lowest value of ​n​ is 1, this gives the lowest possible value of the orbital radius. This is known as the Bohr radius, and it is approximately 0.0529 nanometers. An electron can be no closer to the nucleus than the Bohr radius and still be in a stable orbit.

Each value of ​n​ provides a definite energy at a definite radius known as an energy shell or energy level. In these orbits, the electron does not radiate energy and so does not fall into the nucleus.

Bohr's model is consistent with observations leading to quantum theory such as Einstein's photoelectric effect, matter waves and the existence of photons (although Bohr did not believe in the existence of photons).

The Rydberg formula was known empirically before Bohr's model, but it fits Bohr's description of the energies associated with transitions or jumps between excited states. The energy associated with a given orbital transition is

where ​R_E​ is the Rydberg constant, and ​n_f​ and ​n_i​ are the ​n​ values of the final and initial orbitals, respectively.

Bohr's model gives an incorrect value for the ground state's (lowest energy state's) angular momentum; its model predicts a value of ħ when the true value is known to be zero. The model is also not effective in predicting the energy levels of larger atoms, or atoms with more than one electron. It is most accurate when applied to a hydrogen atom.

The model violates Heisenberg's uncertainty principle in that it considers electrons to have known orbits ​and​ locations. According to the uncertainty principle, these two things cannot be simultaneously known about a quantum particle.

There are also quantum effects that are not explained by the model, such as the Zeeman effect and the existence of fine and hyperfine structure in spectral lines.

Two main atomic models were created before Bohr's. In Dalton's model, an atom was simply a fundamental unit of matter. Electrons were not considered. J.J. Thomson's plum pudding model was an extension of Dalton's, which represented electrons as being embedded in a solid like raisins in a pudding.

Schrödinger's electron cloud model came after Bohr's and represented the electrons as being spherical clouds of probability that grow denser near the nucleus.