  Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$.
Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the
number of different non-zero ranks that occur among the elements of $M$. Our
aim is to obtain an upper bound for $\dim M$ in terms of $r$ and $n$ under
various hypotheses. As a sample of what we prove, we mention the following.
Suppose that $m$ is the largest integer that occurs as the rank of an element
of $M$. Then if $m\leq \lceil n/2\rceil$ and $|K|\geq m+1$, we have $\dim M\leq
rn$. The case $r=1$ corresponds to a constant rank space and it is conjectured
that $\dim M\leq n$ when $M$ is a constant rank $m$ space and $|K|\geq m+1$. We
prove that the dimension bound for a constant rank $m$ space $M$ holds provided
$|K|\geq m+1$ and either $K$ is finite or $K$ has characteristic different from
2 and $M$ consists of symmetric forms. In general, we show that if $M$ is a
constant rank $m$ subspace and $|K|\geq m+1$, then $\dim M\leq \max\,(n,2m-1)$.
We also provide more detailed results about constant rank subspaces over finite
fields, especially subspaces of alternating or symmetric bilinear forms.
