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• Existence and exponential boundedness of solution of fractional neutral delay differential equations are investigated. • Explicit solutions of linear equations are derived in terms of generalized Mittag–Leffler functions. • Stability properties are analyzed. • Numerical simulations illustrate the asymptotic behavior. This paper discusses some properties of solutions to fractional neutral delay differential equations. By combining a new weighted norm, the Banach fixed point theorem and an elegant technique for extending solutions, results on existence, uniqueness, and growth rate of global solutions under a mild Lipschitz continuous condition of the vector field are first established. Be means of the Laplace transform the solution of some delay fractional neutral differential equations are derived in terms of three-parameter Mittag–Leffler functions; their stability properties are hence studied by using use Rouché's theorem to describe the position of poles of the characteristic polynomials and the final value theorem to detect the asymptotic behavior. By means of numerical simulations the theoretical findings on the asymptotic behavior are verified. [ABSTRACT FROM AUTHOR]