  Let $B_n$ denote the classical braid group on $n$ strands and let the {\em
mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for
which the first $m$ strands form the identity braid. Let
$B_{m,\infty}=\cup_nB_{m,n}$. We will describe explicit algebraic moves on
$B_{m,\infty}$ such that equivalence classes under these moves classify
oriented links up to isotopy in a link complement or in a closed, connected,
oriented 3--manifold. The moves depend on a fixed link representing the
manifold in $S^3$. More precisely, for link complements the moves are: the two
familiar moves of the classical Markov equivalence together with {\em `twisted'
conjugation} by certain loops $a_i$. This means premultiplication by
${a_i}^{-1}$ and postmultiplication by a `combed' version of $a_i$. For closed
3--manifolds there is an additional set of {\it `combed' band moves} which
correspond to sliding moves over the surgery link. The main tool in the proofs
is the one-move Markov Theorem using {\it $L$--moves} \cite{LR} (adding in-box
crossings). The resulting algebraic classification is a direct extension of the
classical Markov Theorem that classifies links in $S^3$ up to isotopy, and
potentially leads to powerful new link invariants, which have been explored in
special cases by the first author. It also provides a controlled range of
isotopy moves, useful for studying skein modules of 3--manifolds.
