The theory of quaternionic linear operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis. A major obstacle to develop mathematical tools in this field was due to a lack of a suitable notion of spectrum. The introduction of the notion of S-spectrum in 2006 opened the way to systematic development of the theory. In this talk, we illustrate the basic properties of the S-spectrum, of the so-called S-resolvent operators, of the associated functional calculus and we discuss some applications. In particular, we discuss some results in quaternionic Schur analysis, and we deepen the discussion of some operators and their perturbation. References D. Alpay, F. Colombo, I. Sabadini, Slice Hyperholomorphic Schur analysis, Slice hyperholomorphic Schur analysis. Operator Theory: Advances and Applications, 256. Birkhauser/Springer, Cham, 2016. xii+362. F. Colombo, J. Gantner, D.P. Kimsey, Spectral theory on the S-spectrum for quaternionic operators, Operator Theory: Advances and Applications, 270. Birkhauser/Springer, Cham, 2018. ix+356 pp. F. Colombo, I. Sabadini, D. C. Struppa, Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions, Progress in Mathematics, Vol. 289, Birkhauser, 2011, VI, 222 p.