  In 1952 Lee and Yang proposed the program of analyzing phase transitions in
terms of zeros of partition functions. Linear operators preserving
non-vanishing properties are essential in this program and various contexts in
complex analysis, probability theory, combinatorics, and matrix theory. We
characterize all linear operators on finite or infinite-dimensional spaces of
multivariate polynomials preserving the property of being non-vanishing
whenever the variables are in prescribed open circular domains. In particular,
this solves the higher dimensional counterpart of a long-standing
classification problem originating from classical works of Hermite, Laguerre,
Hurwitz and P\'olya-Schur on univariate polynomials with such properties.
