  We consider the critical behaviour of long-range $O(n)$ models ($n \ge 0$) on
${\mathbb Z}^d$, with interaction that decays with distance $r$ as
$r^{-(d+\alpha)}$, for $\alpha \in (0,2)$. For $n \ge 1$, we study the
$n$-component $|\varphi|^4$ lattice spin model. For $n =0$, we study the weakly
self-avoiding walk via an exact representation as a supersymmetric spin model.
These models have upper critical dimension $d_c=2\alpha$. For dimensions
$d=1,2,3$ and small $\epsilon>0$, we choose $\alpha = \frac 12 (d+\epsilon)$,
so that $d=d_c-\epsilon$ is below the upper critical dimension. For small
$\epsilon$ and weak coupling, to order $\epsilon$ we prove existence of and
compute the values of the critical exponent $\gamma$ for the susceptibility
(for $n \ge 0$) and the critical exponent $\alpha_H$ for the specific heat (for
$n \ge 1$). For the susceptibility, $\gamma = 1 + \frac{n+2}{n+8} \frac
\epsilon\alpha + O(\epsilon^2)$, and a similar result is proved for the
specific heat. Expansion in $\epsilon$ for such long-range models was first
carried out in the physics literature in 1972. Our proof adapts and applies a
rigorous renormalisation group method developed in previous papers with
Bauerschmidt and Brydges for the nearest-neighbour models in the critical
dimension $d=4$, and is based on the construction of a non-Gaussian
renormalisation group fixed point. Some aspects of the method simplify below
the upper critical dimension, while some require different treatment, and new
ideas and techniques with potential future application are introduced.
