A relativistic version of the Schrödinger equation.

Correctly describes spin 0 particles.

The most memorable version of the equation can be written as shown at Section "Klein-Gordon equation in Einstein notation" with Einstein notation and Planck units:

\partial_{i} \partial^{i} ψ - m^{2} ψ = 0

Has some issues which are solved by the Dirac equation:

it has a second time derivative of the wave function. Therefore, to solve it we must specify not only the initial value of the wave equation, but also the derivative of the wave equation,
As mentioned at Advanced quantum mechanics by Freeman Dyson (1951) and further clarified at: physics.stackexchange.com/questions/340023/cant-the-negative-probabilities-of-klein-gordon-equation-be-avoided, this would lead to negative probabilities.
the modulus of the wave function is not constant and therefore not always one, and therefore cannot be interpreted as a probability density anymore
since we are working with the square of the energy, we have both positive and negative value solutions. This is also a features of the Dirac equation however.

Bibliography:

 Ancestors

Klein-Gordon equation