Tissue engineering is nowadays a powerful tool to restore damaged tissues and recover their normal functionality. Advantages over other current methods are well established, although a continuous evolution is still necessary to improve the final performance and the range of applications. Trends are nowadays focused on the development of multifunctional scaffolds with hierarchical structures and the capability to render a sustained delivery of bioactive molecules under an appropriate stimulus. Nanocomposites incorporating hydroxyapatite nanoparticles (HAp NPs) have a predominant role in bone tissue regeneration due to their high capacity to enhance osteoinduction, osteoconduction, and osteointegration, as well as their encapsulation efficiency and protection capability of bioactive agents. Selection of appropriated polymeric matrices is fundamental and consequently great efforts have been invested to increase the range of properties of available materials through copolymerization, blending, or combining structures constituted by different materials. Scaffolds can be obtained from different processes that differ in characteristics, such as texture or porosity. Probably, electrospinning has the greater relevance, since the obtained nanofiber membranes have a great similarity with the extracellular matrix and, in addition, they can easily incorporate functional and bioactive compounds. Coaxial and emulsion electrospinning processes appear ideal to generate complex systems able to incorporate highly different agents. The present review is mainly focused on the recent works performed with Hap-loaded scaffolds having at least one structural layer composed of core/shell nanofibers.