Reconfigurable antennas have gained tremendous interest owing to their multifunctional capabilities while adhering to minimalistic space requirements in ever-shrinking electronics platforms and devices. A stark increase in demand for flexible and conformal antennas in modern and emerging unobtrusive and space-limited electronic systems has led to the development of the flexible and conformal reconfigurable antennas era. Flexible and conformal antennas rely on non-conventional materials and realization approaches, and thus, despite the mature knowledge available for rigid reconfigurable antennas, conventional reconfigurable techniques are not translated to a flexible domain in a straight forward manner. There are notable challenges associated with integration of reconfiguration elements such as switches, mechanical stability of the overall reconfigurable antenna, and the electronic robustness of the resulting devices when exposed to folding of sustained bending operations. This paper reviews various approaches demonstrated thus far, to realize flexible reconfigurable antennas, categorizing them on the basis of reconfiguration attributes, i.e., frequency, pattern, polarization, or a combination of these characteristics. The challenges associated with development and characterization of flexible and conformal reconfigurable antennas, the strengths and limitations of available methods are reviewed considering the progress in recent years, and open challenges for the future research are identified.