We study the continuous and discrete versions of the Markus-Yamabe Conjecture
for polynomial vector fields in $ \mathbb{R}^3 $ of the form $ X = \lambda \, I
+ H $, where $ \lambda $ is a real number, I the identity map, and H a map with
nilpotent Jacobian matrix $ JH $. We distinguish the cases when the rows of $ J
H $ are linearly dependent over $ \mathbb{R} $ and when they are linearly
independent over $ \mathbb{R} $.

In the dependent continuous case, we give a polynomial family of
counterexamples to the Markus-Yamabe conjecture which contains and generalizes
that of Cima-Gasull-Ma\~nosas. Furthermore, we construct a new class of
polynomial vector fields in $\mathbb{R}^3$ having the origin as a global
attractor. We also find non--linearly triangularizable vector fields $ X $ for
which the origin is a global attractor for both the continuous and the discrete
dynamical systems generated by $ X $.

In the independent continuous case, we present a family of vector fields that
have orbits escaping to infinity.