  Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to an
affine IFS $\Phi$ and a positive probability vector $p$. Suppose that the maps
in $\Phi$ do not have a common fixed point, and that standard irreducibility
and proximality assumptions are satisfied by their linear parts. We show that
$\dim\mu$ is equal to the Lyapunov dimension $\dim_{L}(\Phi,p)$ whenever $d=3$
and $\Phi$ satisfies the strong separation condition (or the milder strong open
set condition). This follows from a general criteria ensuring
$\dim\mu=\min\{d,\dim_{L}(\Phi,p)\}$, from which earlier results in the planar
case also follow. Additionally, we prove that $\dim\mu=d$ whenever $\Phi$ is
Diophantine (which holds e.g. when $\Phi$ is defined by algebraic parameters)
and the entropy of the random walk generated by $\Phi$ and $p$ is at least
$(\chi_{1}-\chi_{d})\frac{(d-1)(d-2)}{2}-\sum_{k=1}^{d}\chi_{k}$, where
$0>\chi_{1}\ge...\ge\chi_{d}$ are the Lyapunov exponents. We also obtain
results regarding the dimension of orthogonal projections of $\mu$.
