  Let $X$ be a compact smooth manifold with boundary. In this article, we study
the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called
boundary generic and traversally generic vector fields on $X$ and the place
they occupy in the space $\mathcal V(X)$ of all fields (see Theorems
\ref{th3.4} and Theorem \ref{th3.5}). The definitions of boundary generic and
traversally generic vector fields $v$ are inspired by some classical notions
from the singularity theory of smooth Bordman maps \cite{Bo}. Like in that
theory (cf. \cite{Morin}), we establish local versal algebraic models for the
way a sheaf of $v$-trajectories interacts with the boundary $\d X$. For fields
from the space $\mathcal V^\ddagger(X)$, the finite list of such models depends
only on $\dim(X)$; as a result, it is universal for all equidimensional
manifolds. In specially adjusted coordinates, the boundary and the $v$-flow
acquire descriptions in terms of universal deformations of real polynomials
whose degrees do not exceed $2\cdot \dim(X)$.
