A set $\Lambda \subset R^d$ is called self-affine if there is a finite list $F:=\{f_1(x)=A_1\cdot x + t_1, \dots ,f_m(x)=A_m\cdot x + t_m\}$ of contracting affine transformations such that $\Lambda $ can be presented as the union of its on affine copies with these affine maps: $\Lambda =\bigcup_{i=1}^{n}f_i(\Lambda )$. In this case we say that $\Lambda $ is the attractor of the self-affine Iterated Function System $F$. Falconer-Solomyak Theorem says that in those cases when all the linear parts contract stronger than $1/2$ for almost all $t_1, \dots ,t_m \in R^{m\cdot d}$ the Hausdorff dimension of $\Lambda $ is equal to the so called singularity dimension, which is actually the number that we obtain from the definition of the Hausdorff dimension if we use the most natural system of covers. The same is known to be false if we drop the assumption: $\forall i,\ \|A_i\|<1/2$. Actually we know very little about the dimension theory of self-affine IFS. To make some progress in 2006 we (T. Jordan, M. Pollicott, K. Simon) introduced a random IFS which is obtained from the deterministic self-affine IFS in such a way that at each application of the system we add an arbitrary small random error independently. We obtain a random IFS that is called almost self-affine system. In this talk I give a review about the dimension theory of almost self-affine sets andmeasures. I also review some recent results about the multifractal analysis of some almost self-affine measures.