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We consider transverse oscillations of coronal loops first observed by TRACE on 14 July 1998. We mainly concentrate on theory of these oscillations, although relevant observational results are also discussed. The transverse coronal loop oscillations were interpreted as standing fast kink waves in magnetic flux tubes. We start the review from the discussion of theory of kink waves in a homogeneous straight magnetic cylinder. Then we consider the effects of stratification, loop expansion and curvature, the twist of magnetic field lines, and non-circular cross-section. An important property of observed transverse coronal loop oscillations is their fast damping. We present the theory of damping due to resonant absorption. First we describe the analytical results obtained with the use of thin transitional layer approximation. Then we compare them with numerical results obtained for arbitrary density variation inside the tube. Finally we discuss the implication of theoretical results for coronal seismology. [ABSTRACT FROM AUTHOR]