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The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them. Some historyThis is an expository paper, but because of the demands of the journal, we had to restrict very significantly the number of references. We apologize to the authors of all the papers that are connected with the subjects considered here, but had to be omitted.The property of additivity of a mapping is very important (not only in mathematics) and can be described by the following Cauchy functional equationwhere f is a mapping between semigroups endowed with binary operations denoted by + (we abuse the notation and we use the same symbol for operations in two different structures). Every mapping satisfying equation (1.1) is said to be

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We investigate some inequalities connected with the Hyers-Ulam stability of three functional equations, which have a solution of the form ϕ = a + q, where a is an additive mapping and q is a quadratic one.

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