  Six-dimensional superconformal field theories (SCFTs) have an atomic
classification in terms of elementary building blocks, conformal systems that
generalize matter and can be fused together to form all known 6d SCFTs in terms
of generalized 6d quivers. It is therefore natural to ask whether 5d SCFTs can
be organized in a similar manner, as the outcome of fusions of certain
elementary building blocks, which we call 5d conformal matter theories. In this
project we begin exploring this idea and we give a systematic construction of
5d generalized ``bifundamental'' SCFTs, building from geometric engineering
techniques in M-theory. In particular, we find several examples of $(\mathfrak
{e}_6,\mathfrak {e}_6)$, $(\mathfrak {e}_7,\mathfrak {e}_7)$ and $(\mathfrak
{e}_8,\mathfrak {e}_8)$ 5d bifundamental SCFTs beyond the ones arising from
(elementary) KK reductions of the 6d conformal matter theories. We show that
these can be fused together giving rise to 5d SCFTs captured by 5d generalized
linear quivers with exceptional gauge groups as nodes, and links given by 5d
conformal matter. As a first application of these models we uncover a large
class of novel 5d dualites, that generalize the well-known fiber/base dualities
outside the toric realm.
