  Observations of astrophysical transients have brought many novel discoveries
and provided new insights into physical processes at work under extreme
conditions in the Universe. Multi-wavelength and multi-messenger observations
of variable objects require dedicated procedures and follow-up systems capable
of digesting and reacting to external alerts to execute coordinated follow-up
campaigns. The main functions of such follow-up systems are the processing,
filtering, and ranking of the incoming alerts, the fully automated rapid
execution of the observations according to an observation strategy tailored to
the instrument, and real-time data analysis with feedback to the operators and
other instruments. H.E.S.S. has been searching for transient phenomena since
its inauguration in 2003. In this paper, we describe the transients follow-up
system of H.E.S.S. which became operational in 2016. The system allows H.E.S.S.
to conduct a more versatile, optimised, and largely autonomous transient
follow-up program, combining all major functionalities in one systematic
approach. We describe the design, central functionalities, and interfaces of
the follow-up system in general and its three main components in detail: the
Target of Opportunity (ToO) alert system, the data acquisition and central
control system, and the real-time analysis. We highlight architectural
decisions and features that enable fully automatic ToO follow-up and indicate
key performance metrics of the sub-systems. We discuss the system's
capabilities and highlight the need for a fine-tuned interplay of the different
sub-systems in order to react quickly and reliably. Lessons learned from the
development, integration, and operation of the follow-up system are reviewed in
light of new and large science infrastructures and associated challenges in
this exciting new era of inter-operable astronomy.
