  Totally real immersions $f$ of a closed real surface $\Sigma$ in an almost
complex surface $M$ are completely classified, up to homotopy through totally
real immersions, by suitably defined homotopy classes $\frak{M}(f)$ of mappings
from $\Sigma$ into a specific real 5-manifold $E(M)$, while $\frak{M}(f)$
themselves are subject to a single cohomology constraint. This follows from
Gromov's observation that totally real immersions satisfy the h-principle. For
the receiving complex surfaces $C^2$, $CP^1\times CP^1$, $CP^2$ and $CP^2 #
m\bar{CP^2}$, $m=1,2,...,7$, and all $\Sigma$ (or, $CP^2 # 8\bar{CP^2}$ and all
orientable $\Sigma$), we illustrate the above nonconstructive result with
explicit examples of immersions realizing all possible equivalence classes. We
also determine which equivalence classes contain totally real embeddings, and
provide examples of such embeddings for all classes that contain them.
