  This article mainly aims to overview the recent efforts on developing
algebraic geometry for an arbitrary compact almost complex manifold.
  We review the results obtained by the guiding philosophy that a statement for
smooth maps between smooth manifolds in terms of Ren\'e Thom's transversality
should also have its counterpart in pseudoholomorphic setting without requiring
genericity of the almost complex structures. These include intersection of
compact almost complex submanifolds, structure of pseudoholomorphic maps, zero
locus of certain harmonic forms, and eigenvalues of Laplacian. In addition to
reviewing the compact manifolds situation, we also extend these results to
orbifolds and non-compact manifolds. Motivations, methodologies, applications,
and further directions are discussed.
  The structural results on the pseudoholomorphic maps lead to a notion of
birational morphism between almost complex manifolds. This motivates the study
of various birational invariants, including Kodaira dimensions and plurigenera,
in this setting. Some other aspects of almost complex algebraic geometry in
dimension $4$ are also reviewed. These include cones of (co)homology classes
and subvarieties in spherical classes.
