  We study locally harmonic maps between a Riemann surface or Lorentz surface
$M$ and a Riemann surface or Lorentz surface $N$. {All four cases are studied
in a unified way}. All four cases are written using a unified formalism.
Therefore solutions to the harmonic map problem can be studied in a unified
way.
  Harmonic maps between pseudo-Riemannian surfaces are classified by the
classification of the solutions of a generalized sine-Gordon equation. We then
study the one-soliton solutions of this equation and we find the corresponding
harmonic maps in a unified way.
  Next, we discuss a B\"acklund transformation of the harmonic map equations
that provides a connection between the solutions of two sine-Gordon type
equations. Finally, we give an example of a harmonic map that is constructed by
the use of a B\"acklund transformation.
