  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 $\mathcal{H}^{t}(g(K))=0$.
