  This paper is concerned with restricted families of projections in
$\mathbb{R}^{3}$. Let $K \subset \mathbb{R}^{3}$ be a Borel set with Hausdorff
dimension $\dim K = s > 1$. If $\mathcal{G}$ is a smooth and sufficiently
well-curved one-dimensional family of two-dimensional subspaces, the main
result states that there exists $\sigma(s) > 1$ such that $\dim \pi_{V}(K) \geq
\sigma(s)$ for almost all $V \in \mathcal{G}$. A similar result is obtained for
some specific families of one-dimensional subspaces.
