  This paper presents the current status on modularity of Calabi-Yau varieties
since the last update in 2003. We will focus on Calabi-Yau varieties of
dimension at most three. Here modularity refers to at least two different
types: arithmetic modularity and geometric modularity. These will include: (1)
the modularity (automorphy) of Galois representations of Calabi-Yau varieties
(or motives) defined over Q or number fields, (2) the modularity of solutions
of Picard--Fuchs differential equations of families of Calabi-Yau varieties,
and mirror maps (mirror moonshine), (3) the modularity of generating functions
of invariants counting certain quantities on Calabi-Yau varieties, and (4) the
modularity of moduli for families of Calabi-Yau varieties.
