  It is classically known that the only zero mean curvature entire graphs in
the Euclidean 3-space are planes, by Bernstein's theorem. A surface in
Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it
changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$,
Osamu Kobayashi found two zero mean curvature entire graphs of mixed type that
are not planes. As far as the authors know, these two examples were the only
known examples of entire zero mean curvature graphs of mixed type without
singularities. In this paper, we construct several families of real analytic
zero mean curvature entire graphs of mixed type in Lorentz-Minkowski $3$-space.
The entire graphs mentioned above lie in one of these classes.
