  Fixed points for scalar theories in $4-\varepsilon$, $6-\varepsilon$ and
$3-\varepsilon$ dimensions are discussed. It is shown how a large range of
known fixed points for the four dimensional case can be obtained by using a
general framework with two couplings. The original maximal symmetry, $O(N)$, is
broken to various subgroups, both discrete and continuous. A similar discussion
is applied to the six dimensional case. Perturbative applications of the
$a$-theorem are used to help classify potential fixed points. At lowest order
in the $\varepsilon$-expansion it is shown that at fixed points there is a
lower bound for $a$ which is saturated at bifurcation points.
