  We investigate in this work some situations where it is possible to estimate
or determine the upper and the lower $q$-generalized fractal dimensions
$D^{\pm}_{\mu}(q)$, $q\in\mathbb{R}$, of invariant measures associated with
continuous transformations over compact metric spaces. In particular, we
present an alternative proof of Young's Theorem~\cite{Young} for the
generalized fractal dimensions of the Bowen-Margulis measure associated with a
$C^{1+\alpha}$-Axiom A system over a two-dimensional compact Riemannian
manifold $M$. We also present estimates for the generalized fractal dimensions
of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually,
in terms of its metric entropy.
  Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we
show that the set of invariant measures such that $D_\mu^+(q)=0$ ($q\ge 1$),
under a hyperbolic metric, is generic (taking into account the weak topology).
We also show that for each $s\in [0,1)$, $D^{+}_{\mu}(s)$ is bounded above, up
to a constant, by the topological entropy, also under a hyperbolic metric.
  Finally, we show that, for some dynamical systems, the metric entropy of an
invariant measure is typically zero, settling a conjecture posed by Sigmund
in~\cite{Sigmund1974} for Lipschitz transformations which satisfy the
specification property.
