  Interface localised interactions are studied for multiscalar universality
classes accessible with the perturbative $\varepsilon$ expansion in
$4-\varepsilon$ dimensions. The associated beta functions at one loop and
partially at two loops are derived, and a wide variety of interface conformal
field theories (CFTs) is found, even in cases where the bulk universality class
is free or as simple as the Wilson-Fisher description of the $O(N)$ model. For
up to three scalar fields in the bulk, interface fixed points are classified
for all bulk universality classes encountered in this case. Numerical results
are obtained for interface CFTs that exist for larger numbers of multiscalar
fields. Our analytic and numerical results indicate the existence of a vast
space of interface CFTs, much larger than the space of defect CFTs found for
line and surface defect deformations of multiscalar models in $4-\varepsilon$
dimensions. In this vast space, stable interfaces found for free and $O(N)$
bulks belong to the $F_4$ family, with global symmetries $SO(3), SU(3), Sp(6)$
and $F_4$, realised with $N=5,8,16,24$ scalar fields, respectively.
