  Hilbert's Nullstellensatz is one of the most fundamental correspondences
between algebra and geometry, and has inspired a plethora of noncommutative
analogs. In last two decades, there has been an increased interest in
understanding vanishing sets of polynomials in several matrix variables without
restricting the matrix size, prompted by developments in noncommutative
function theory, control systems, operator algebras, and quantum information
theory. The emerging results vary according to the interpretation of what
vanishing means. For example, given a collection of noncommutative polynomials,
one can consider all matrix tuples at which the values of these polynomials are
all zero, singular, have common kernel, or have zero trace. This survey reviews
Nullstellens\"atze for the above types of vanishing sets, and identifies their
structural counterparts in the free algebra.
