In this monograph the author presents explicit conditions for the exponential, absolute  and  input-to-state stabilities -- including solution estimates -- of certain types of functional differential equations. The main methodology used is based on a combination of recent norm estimates for matrix-valued functions, comprising the generalized Bohl-Perron principle, together with its integral version and the positivity of fundamental solutions. A significant part of the book is especially devoted  to the solution of the generalized Aizerman problem.