  We study the geometry of the Fano schemes
$\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ of the projective variety
$\mathrm{SD}_n^r$ defined by the $r\times r$ minors of a symmetric $n\times n$
matrix filled with indeterminates. These schemes are fine moduli spaces
parameterizing $(k+1)$-dimensional linear spaces of symmetric matrices of rank
less than $r$. We prove that the schemes
$\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ can have generically non-reduced
components, characterize their irreducibility and connectedness, and give
results on their smoothness. Our approach to connectedness also applies to Fano
schemes of rectangular matrices as well as alternating matrices and answers a
question of Ilten and Chan. Furthermore, we give a complete description of
$\mathrm{\textbf{F}}_{1}(\mathrm{SD}_n^r)$ and show that when $r=n$, the Fano
schemes of lines have the expected dimension. As an application, we provide
geometric arguments for several previous results concerning spaces of symmetric
matrices of bounded rank.
