  Various families of charge-one vortex solitons in two-dimensional periodic
media are reported. These vortices reside either in the semi-infinite gap or
higher band gaps of the media. For both Kerr and saturable nonlinearities
(either focusing or defocusing), infinite vortex families are found. All these
families do not bifurcate from Bloch bands; rather, they turn around before
reaching edges of Bloch bands. It is further revealed that vortices with
drastically different topological shapes can belong to the same vortex family,
which is quite surprising.
