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ABSTRACT: In this talk we aim to describe the rich relation between 1-dimensional algebraic torus actions on projective varieties and birational geometry. This seminar is divided in two parts. In the first one we give a provide a general introduction to the subject: we describe for instance the Bialynicki-Birula decomposition, GIT quotients of a polarized pair by a C*-action and the recent results obtained by G. Occhetta, E. Romano, L. E. Solà Conde and J. Wisniewski. The second part of the talk is based on joint works with E. Romano, G. Occhetta, L. Sola Conde and S. Urbinati. Given a birational map among projective varieties, we introduce a projective algebraic version of the cobordism notion coming from Morse theory, which we call geometric realization. We show how to construct geometric realizations of maps which are small modifications of dream type, that is birational maps which are isomorphism in codimension 1 associated to a finitely generated multisection ring.