The paper defines, with the help of generalized topologies and generalized neighbourhood systems, two kinds of generalized
continuity, by giving in this way a general form to various concepts discussed in the literature. In particular, generalized
continuity admits a characterization furnishing a known charcterization of θ-continuous maps.
We study the relationship between the product and other basic operations (namely σ, π, α and β) of generalized topologies. Also we discuss the connectedness, generalized connectedness and compactness of products of generalized
topologies. It is proved that the connectedness and compactness are preserved under the product of generalized topologies,
which shows that the definition of product of generalized topologies is quite reasonable.