  We investigate the critical behavior that d-dimensional systems with
short-range forces and a n-component order parameter exhibit at Lifshitz points
whose wave-vector instability occurs in a m-dimensional isotropic subspace of
${\mathbb R}^d$. Utilizing dimensional regularization and minimal subtraction
of poles in $d=4+{m\over 2}-\epsilon$ dimensions, we carry out a two-loop
renormalization-group (RG) analysis of the field-theory models representing the
corresponding universality classes. This gives the beta function $\beta_u(u)$
to third order, and the required renormalization factors as well as the
associated RG exponent functions to second order, in u. The coefficients of
these series are reduced to m-dependent expressions involving single integrals,
which for general (not necessarily integer) values of $m\in (0,8)$ can be
computed numerically, and for special values of m analytically. The $\epsilon$
expansions of the critical exponents $\eta_{l2}$, $\eta_{l4}$, $\nu_{l2}$,
$\nu_{l4}$, the wave-vector exponent $\beta_q$, and the correction-to-scaling
exponent are obtained to order $\epsilon^2$. These are used to estimate their
values for d=3. The obtained series expansions are shown to encompass both
isotropic limits m=0 and m=d.
