  In this paper, we first study biplanes $\mathcal{D}$ with parameters
$(v,k,2)$, where the block size $k\in\{13,16\}$. These are the smallest
parameter values for which a classification is not available. We show that if
$k=13$, then either $\mathcal{D}$ is the Aschbacher biplane or its dual, or
$Aut(\mathcal{D})$ is a subgroup of the cyclic group of order $3$. In the case
where $k=16$, we prove that $|Aut(\mathcal{D})|$ divides $2^{7}\cdot 3^{2}\cdot
5\cdot 7\cdot 11\cdot 13$. We also provide an example of a biplane with
parameters $(16,6,2)$ with a flag-transitive and point-primitive subgroup of
automorphisms preserving a homogeneous cartesian decomposition. This motivated
us to study biplanes with point-primitive automorphism groups preserving a
cartesian decomposition. We prove that such an automorphism group is either of
affine type (as in the example), or twisted wreath type.
