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We show that if $K$ is a self-similar $1$-set that is not contained in a line and either satisfies the strong separation condition or is defined via homotheties then there are at most finitely many lines through the origin such that the projection of $K$ onto them is an interval.

We show that for the attractor $(K_{1},\dots,K_{q})$ of a graph directed iterated function system, for each $1\leq j\leq q$ and $\varepsilon>0$ there exits a self-similar set $K\subseteq K_{j}$ that satisfies the strong separation condition and $\dim_{H}K_{j}-\varepsilon<\dim_{H}K$. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of $K$. Using this property as a `black box' we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.

We are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph-directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. For example, it fails in general for self-conformal sets, self-affine sets and Julia sets. We also give applications of our results concerning Ahlfors regularity. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and $\delta$-approximate packing pre-measure coincide for sufficiently small $\delta>0$., Comment: 21 pages. This version includes applications concerning the weak separation property and Ahlfors regularity. To appear in Journal of Fractal Geometry

We investigate how the Hausdorff dimension and measure of a self-similar set $K\subseteq\mathbb{R}^{d}$ behave under linear images. This depends on the nature of the group $\mathcal{T}$ generated by the orthogonal parts of the defining maps of $K$. We show that if $\mathcal{T}$ is finite then every linear image of $K$ is a graph directed attractor and there exists at least one projection of $K$ such that the dimension drops under the image of the projection. In general, with no restrictions on $\mathcal{T}$ we establish that $\mathcal{H}^{t}(L\circ O(K))=\mathcal{H}^{t}(L(K))$ for every element $O$ of the closure of $\mathcal{T}$, where $L$ is a linear map and $t=\dim_{H}K$. We also prove that for disjoint subsets $A$ and \textbf{$B$} of $K$ we have that $\mathcal{H}^{t}(L(A)\cap L(B))=0$. Hochman and Shmerkin showed that if $\mathcal{T}$ is dense in $SO(d,\mathbb{R})$ and the strong separation condition is satisfied then $\dim_{H}(g(K))=\min\{\dim_{H}K,l\} $ where $g$ is a continuously differentiable map of rank $l$. We deduce the same result without any separation condition and we generalize a result of Ero$\breve{\mathrm{g}}$lu by obtaining that $\mathcal{H}^{t}(g(K))=0$.