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The paper deals with the Sturm-Liouville operator $$Ly=-y^{\prime\prime}+q(x)y,\qquad x\in\lbrack0,1], $$ generated in the space$L_{2}=L_{2}[0,1]$ by periodic or antiperiodic boundary conditions. Severaltheorems on Riesz basis property of the root functions of the operator $L$ areproved. One of the main results is the following. \textsl{Let $q$ belong toSobolev space $W_{1}^{p}[0,1]$ with some integer $p\geq0$ and satisfy theconditions $q^{(k)}(0)=q^{(k)}(1)=0$ for $0\leq k\leq s-1$, where s}$\leq p.$\textsl{Let the functions $Q$ and $S$ be defined by the equalities$Q(x)=\int_{0}^{x}q(t) dt, S(x)=Q^{2}(x)$ and let $q_{n}%, Q_{n},S_{n}$ be theFourier coefficients of $q,Q,S$ with respect to the trigonometric system$\{e^{2\pi inx}\}_{-\infty}^{\infty}$. Assume that the sequence$q_{2n}-S_{2n}+2Q_{0}Q_{2n}$ decreases not faster than the powers $n^{-s-2}$.Then the system of eigen and associated functions of the operator $L$ generatedby periodic boundary conditions forms a Riesz basis in the space $L_{2}[0,1]$(provided that the eigenfunctions are normalized) if and only if the condition$$ q_{2n}-S_{2n}+Q_{0}Q_{2n}\asymp q_{-2n}-S_{-2n}+2Q_{0}Q_{-2n},\quad n>1, $$holds.

On the Riesz Basis Property of the Eigen- and Associated Functions of Periodic and Antiperiodic Sturm-Liouville Problems