Analytical forms of the rectilinear equation of motion and energy equation of particles, droplets or bubbles have been developed for very low Reynolds and Peclet numbers. Some of the early work on the two equations is briefly explored and recent advances are presented in more detail. Particular emphasis is placed on the analogies and similarities between the momentum and energy equations, and the ways the similarities have been utilized in practice. The creeping Bow assumption, oil which most of the known analytical forms are based, is critically examined. The semiempirical and empirical versions of the momentum and energy equations, which are widely used in engineering practice, are also presented, as well as a numerical method to deal with the history terms. Implicit assumptions on the use of the empirical equations are exposed. An erroneous result pertaining to the droplet hows, and a paradox related to the typical history terms are examined, and their rectification is pointed out. Recent results on the motion and heat transfer al finite Reynolds and Peclet numbers are also exposed. Tn this case, the momentum and thermal wakes around the particle play an important role in the momentum and energy transport process.