  In recent years, bright soliton-like structures composed of gaseous
Bose-Einstein condensates have been generated at ultracold temperature. The
experimental capacity to precisely engineer the nonlinearity and potential
landscape experienced by these solitary waves offers an attractive platform for
fundamental study of solitonic structures. The presence of three spatial
dimensions and trapping implies that these are strictly distinct objects to the
true soliton solutions. Working within the zero-temperature mean-field
description, we explore the solutions and stability of bright solitary waves,
as well as their interactions. Emphasis is placed on elucidating their
similarities and differences to the true bright soliton. The rich behaviour
introduced in the bright solitary waves includes the collapse instability and
symmetry-breaking collisions. We review the experimental formation and
observation of bright solitary matter waves to date, and compare to theoretical
predictions. Finally we discuss the current state-of-the-art of this area,
including beyond-mean-field descriptions, exotic bright solitary waves, and
proposals to exploit bright solitary waves in interferometry and as surface
probes.
