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Let X be a topological space and let T be a bounded operator on X. We say that T is hypercyclic if T admits a dense orbit, namely if there exists a vector x ∈ X, called a hypercyclic vector for T, such that {T^nx; n ≥ 0} is dense in X. We shall denote by HC(T) the set of hypercyclic vectors for T. It is known that, provided HC(T) is nonempty, then it has some nice topological and algebraic properties. For instance, HC(T)∪ {0} always contains a dense subspace, and there are nice criteria for the existence of a closed infinite-dimensional subspace in it. When moreover X is an algebra, it is natural to study whether HC(T) contains a nontrivial algebra. In this talk, we will explain some recent (negative and positive) results on this problem.