  We consider ideals generated by general sets of $m$-minors of an $m\times
n$-matrix of indeterminates. The generators are identified with the facets of
an $(m-1)$-dimensional pure simplicial complex. The ideal generated by the
minors corresponding to the facets of such a complex is called a determinantal
facet ideal. Given a pure simplicial complex $\Delta$, we discuss the question
when the generating minors of its determinantal facet ideal $J_\Delta$ form a
Gr\"obner basis and when $J_\Delta$ is a prime ideal.
