Human-object interaction is of great relevance for robots to operate in human environments. However, state-of-the-art robotic hands are far from replicating humans skills. It is, therefore, essential to study how humans use their hands to develop similar robotic capabilities. This article presents a deep dive into hand-object interaction and human demonstrations, highlighting the main challenges in this research area and suggesting desirable future developments. To this extent, the article presents a general definition of the hand-object interaction problem together with a concise review for each of the main subproblems involved, namely: sensing, perception, and learning. Furthermore, the article discusses the interplay between these subproblems and describes how their interaction in learning from demonstration contributes to the success of robot manipulation. In this way, the article provides a broad overview of the interdisciplinary approaches necessary for a robotic system to learn new manipulation skills by observing human behavior in the real world.