  We discuss in Minkowski spacetime the differences between the concepts of
constant proper $n$-acceleration and of vanishing $(n+1)$-acceleration. By
$n$-acceleration we essentially mean the higher order time derivatives of the
position vector of the trajectory of a point particle, adapted to Minkowski
spacetime or eventually to curved spacetime. The $2$-acceleration is known as
the Jerk, the $3$-acceleration as the Snap, etc. As for the concept of {\sl
proper} $n$-acceleration we give a specific definition involving the
instantaneous comoving frame of the observer and we discuss, in such framework,
the difficulties in finding a characterization of this notion as a Lorentz
invariant statement. We show how the Frenet-Serret formalism helps to address
the problem. In particular we find that our definition of an observer with
constant proper acceleration corresponds to the vanishing of the third
curvature invariant $\kappa_3$ (thus the motion is three dimensional in
Minkowski spacetime) together with the constancy of the first and second
curvature invariants and the restriction $\kappa_2 < \kappa_1$, the particular
case $\kappa_2=0$ being the one commonly referred to in the literature. We
generalize these concepts to curved spacetime, in which the notion of
trajectory in a plane is replaced by the vanishing of the second curvature
invariant $\kappa_2$. Under this condition, the concept of constant proper
$n$-acceleration coincides with that of the vanising of the
$(n+1)$-acceleration and is characterized by the fact that the first curvature
invariant $\kappa_1$ is a $(n-1)$-degree polynomial of proper time. We
illustrate some of our results with examples in Minkowski, de Sitter and
Schwarzschild spacetimes.
