  For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$
such that images of every pairwise distinct $k$ points are affinely (linearly)
independent. Such maps are called affinely (linearly) $k$-regular embeddings.
  We investigate the cohomology obstructions to existence of regular embeddings
and give some new lower bounds on the dimension $m$ as function of $X$ and $k$,
for the cases $X$ is $\mathbb R^n$ or $X$ is an $n$-dimensional manifold. In
the latter case, some nonzero Stiefel--Whitney classes of $X$ help to improve
the bound.
