  This paper contains two results on the dimension and smoothness of radial
projections of sets and measures in Euclidean spaces.
  To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are
non-empty Borel sets with $\dim_{\mathrm{H}} K > 0$. Does the radial projection
of $K$ to some point in $E$ have positive dimension? Not necessarily: $E$ can
be zero-dimensional, or $E$ and $K$ can lie on a common line. I prove that
these are the only obstructions: if $\dim_{\mathrm{H}} E > 0$, and $E$ does not
lie on a line, then there exists a point in $x \in E$ such that the radial
projection $\pi_{x}(K)$ has Hausdorff dimension at least $(\dim_{\mathrm{H}}
K)/2$. Applying the result with $E = K$ gives the following corollary: if $K
\subset \mathbb{R}^{2}$ is Borel set, which does not lie on a line, then the
set of directions spanned by $K$ has Hausdorff dimension at least
$(\dim_{\mathrm{H}} K)/2$.
  For the second result, let $d \geq 2$ and $d - 1 < s < d$. Let $\mu$ be a
compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I
prove that the radial projections of $\mu$ are absolutely continuous with
respect to $\mathcal{H}^{d - 1}$ for every centre in $\mathbb{R}^{d} \setminus
\operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d -
1) - s$. In fact, for $x$ outside an exceptional set as above, the proof shows
that $\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$ for some $p > 1$. The dimension
bound on the exceptional set is sharp.
