  In this paper we define a new family of groups which generalize the {\it
classical braid groups on} $\C $. We denote this family by $\{B_n^m\}_{n \ge
m+1}$ where $n,m \in \N$. The family $\{ B_n^1 \}_{n \in \N}$ is the set of
classical braid groups on $n$ strings. The group $B_n^m$ is the set of motions
of $n$ unordered points in $\C^m$, so that at any time during the motion, each
$m+1$ of the points span the whole of $\C^m$ as an affine space. There is a map
from $B_n^m$ to the symmetric group on $n$ letters. We let $P_n^m$ denote the
kernel of this map. In this paper we are mainly interested in understanding
$P_n^2$. We give a presentation of a group $PL_n$ which maps surjectively onto
$P_n^2$. We also show the surjection $PL_n \to P_n^2$ induces an isomorphism on
first and second integral homology and conjecture that it is an isomorphism. We
then find an infinitesimal presentation of the group $P_n^2$. Finally, we also
consider the analagous groups where points lie in $\P^m$ instead of $\C^m$.
These groups generalize of the classical braid groups on the sphere.
