Companies are gradually developing: 1) complex and/or highly integrated systems including vehicles (as satellites, airplanes, cars, etc.) or equipment (as computers, cell phones, no breaks, etc.) to use under 2) increasingly varied or inhospitable environments, and to survive under 3) increasingly long life cycles and unavoidable changes in staff & facilities & technologies. The overall decision to use (by time, cost, quality, of functions, services, etc.) such end systems under 2 require 4) high Dependability (Reliability, Maintainability, Availability, Correction, Safety, Security, etc.) of them. The overall survival in use (by health monitoring, housekeeping, retrofit, upgrade, etc.) of such end systems under 3 require 5) high Suportability (Maintainability, Adaptability, Availability, Robustness, etc.) of them coupled with the support systems. To meet the requirements and expectations 4 and 5, there is a need to even treat a growing number of faults, arising from 1, 2 and 3 in components, equipment, subsystems or systems used. In particular, health monitoring, fault prognosis and Remaining Useful Life (RUL) prediction have been used to reach 4 and 5 and treat faults in a priori but informed manner. Currently, electromechanical and electrochemical equipment are among the faultiest ones in aerospace and automotive systems. The faults of these equipment can cause decreased performance, operational damage and/or even failures, especially in space systems, since these hardly allow maintenance. So: This paper presents a discussion on algorithms for health monitoring, fault prognosis and RUL prediction of aerospace and automotive equipment. To do that, it: 1) reviews the available literature for health monitoring, fault prognosis and RUL prediction; selects their usual repertoire of faults; 3) highlights some algorithms to treat them; 4) discuss their pros and cons; 5) comment on some cases of electromechanical and electrochemical equipment reported in the available literature. Based on all of this, we expect to show: 1) the adequacy, difficulties and uncertainties in testing and validating such algorithms; and 2) the benefits of health monitoring, fault prognosis and RUL prediction of aerospace and automotive equipment for: a) analysis and anticipation of faults; b) improved dependability, supportability of the respective systems and of the overall decision to use and survival in use of them; c) assistance in sustainable mobility.