In this paper, we classify filiform associative algebras of degree p over a field of characteristic zero. Moreover, over an algebraically closed field of characteristic zero, we also classify filiform nilpotent associative algebras and naturally graded quasi-filiform nilpotent associative algebras, described through the characteristic sequence C(A)=(n-2,1,1)\documentclass[12pt]{minimal}
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\begin{document}$$C({\mathcal {A}})=(n-2,1,1)$$\end{document} or C(A)=(n-2,2)\documentclass[12pt]{minimal}
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\begin{document}$$C({\mathcal {A}})=(n-2,2)$$\end{document}.
