  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. Several
theorems on Riesz basis property of the root functions of the operator $L$ are
proved. One of the main results is the following. \textsl{Let $q$ belong to
Sobolev space $W_{1}^{p}[0,1]$ with some integer $p\geq0$ and satisfy the
conditions $q^{(k)}(0)=q^{(k)}(1)=0$ 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 the
Fourier 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 $L$ generated
by 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.
