The tuned vibration absorber (TVA) is a well-established passive vibration control device for achieving vibration reduction of a primary system subjected to external excitation. This contribution deals with the non-linear dynamics of an adaptive tuned vibration absorber (ATVA) with a shape memory alloy (SMA) element. Initially, a single-degree of freedom oscillator with an SMA element is analyzed showing the general characteristics of its dynamical response. Then, the analysis of an ATVA with an SMA element is carried out. Initially, small amplitude vibrations are considered in such a way that the SMA element does not undergo a stress-induced phase transformation. Under this assumption, the SMA influence is only caused by stiffness changes corresponding to temperature-induced phase transformation. Afterwards, the influence of the hysteretic behavior due to stress-induced phase transformation is considered. A proper constitutive description is employed in order to capture the general thermomechanical aspects of the SMAs. The hysteretic behavior introduces complex characteristics to the system dynamics but also changes the absorber response allowing vibration reduction in different frequency ranges. Numerical simulations establish comparisons of the ATVA results with those obtained from the classical TVA.