  We present a systematic analysis of the N=6 superspace constraints in three
space-time dimensions. The general coupling between vector and scalar
supermultiplets is encoded in an SU(4) tensor $W^i_j$ which is a function of
the matter fields and subject to a set of algebraic and super-differential
relations. We give a genuine N=6 classification for superconformal models with
polynomial interactions and find the known ABJM and ABJ models. We further
study the issue of supersymmetry enhancement to N=8 and the role of monopole
operators in this scenario. To this end we assume the existence of a composite
monopole operator superfield which we use to formulate the additional
supersymmetries as internal symmetries of the N=6 superspace constraints. From
the invariance conditions of these constraints we derive a system of superspace
constraints for the proposed monopole operator superfield. This constraint
system defines the composite monopole operator superfield analogously to the
original N=6 superspace constraints defining the dynamics of the elementary
fields.
