  The weighted triangulation algebras associated to triangulation quivers and
their socle deformations were recently introduced and studied in [15]-[20] and
[2]. These algebras, based on surface triangulations and originated from the
theory of cluster algebras, were also proved to be (with some minor exceptions)
finite-dimensional tame symmetric and periodic algebras of period 4. In this
paper, we introduce a new concept of a generalized triangulation quiver,
extending the notion of a triangulation quiver. Inparticular, it is also shown
that the generalized triangulation quivers can be constructed from
triangulations of orientable surfaces with marked self-foldedtriangles.
Moreover, motivated by the recent results of [28], we define and investigate so
called weighted generalized triangulation algebras associated to generalized
triangulation quivers, which naturally arise from mutations of weighted
triangulation algebras.
