Microfluidics is revolutionizing laboratory methods and biomedical devices, offering new capabilities and instrumentation in multiple areas such as DNA analysis, proteomics, enzymatic analysis, single cell analysis, immunology, point-of-care medicine, personalized medicine, drug delivery, and environmental toxin and pathogen detection. For many applications (e.g., wearable and implantable health monitors, drug delivery devices, and prosthetics) mechanically flexible polymer devices and systems that can conform to the body offer benefits that cannot be achieved using systems based on conventional rigid substrate materials. However, difficulties in implementing active devices and reliable packaging technologies have limited the success of flexible microfluidics. Employing highly compliant materials such as PDMS that are typically employed for prototyping, we review mechanically flexible polymer microfluidic technologies based on free-standing polymer substrates and novel electronic and microfluidic interconnection schemes. Central to these new technologies are hybrid microfabrication methods employing novel nanocomposite polymer materials and devices. We review microfabrication methods using these materials, along with demonstrations of example devices and packaging schemes that employ them. We review these recent developments and place them in the context of the fields of flexible microfluidics and conformable systems, and discuss cross-over applications to conventional rigid-substrate microfluidics.