This paper provides an overview and in-depth analysis of rec ent advances in stability of linear systems with time-varyi ng delays. First, recent developments of a delay convex analys is approach, a reciprocally convex approach and the constru ction of Lyapunov-Krasovskii functionals are reviewed insightful ly. Second, in-depth analysis of the Bessel-Legendre inequ ality and some affine integral inequalities is made, and recent stability r esults are also summarized, including stability criteria f or three cases of a time-varying delay, where information on the bounds of the t ime-varying delay and its derivative is totally known, part ly known and completely unknown, respectively. Third, a number of stabi lity criteria are developed for the above three cases of the t ime-varying delay by employing canonical Bessel-Legendre inequalitie s, together with augmented Lyapunov-Krasovskii functiona ls. It is shown through numerical examples that these stability criteria o utperform some existing results. Finally, several challen ging issues are pointed out to direct the near future research.