Various features that are characteristic of the two-body Dirac equation but not of the one-body Dirac equation are illustrated by means of solvable examples (mainly square-well potentials) in one space dimension. For the Lorentz character of the potential there are three types; vector, scalar, and pseudoscalar. We classify the bound-state solutions as normal and abnormal. As the interaction is adiabatically switched off, the energy of the normal solutions reaches 2m (the sum of the masses of the constituent particles), whereas the energy of the abnormal solutions becomes zero. When the sharp edge of the square-well potential is smeared out, some of the solutions become unnormalizable and hence unacceptable. This leads to a certain restriction on the choice of the potential. The two-body Dirac equation with a finite-range interaction is not exactly covariant. The degree of noncovariance is examined.