We study connections between polynomials which are close to each other, i.e., whose respective coefficients are close in the topology induced by a valuation. This paper consists of both an overview on known root continuity theorems and new results on the subject. We present theorems which have been published over the years, correcting and improving some of their formulations and proofs. We also give a sketch of several approaches to root continuity, such as employing the Newton Polygon and induction on the degree of the polynomial. We study the behavior of the irreducible factors of a given polynomial, the extensions generated by its roots and invariants connected to that polynomial under transition to a second polynomial which is sufficiently close. Further, we present applications of root continuity to the study of valued field extensions and ramification theory, including the theory of the defect.