The paper contains three main results on the algebraic limit cycles for Liénard system in the form
\[ \dot{x}=y,\quad \dot{y}=-f_m(x)y - g_n(x), \]
where \(f_m\) and \(g_n\) are polynomials of degree \(m\) and \(n\), respectively. The first one presents explicit examples of Liénard systems with \(m>1\) and \(n\geq2m+1\) or \(m>2\) and \(n=2m\) which have algebraic limit cycles. The second result is the proof that the Liénard systems for \(m=3\) and \(n=5\) have no algebraic limit cycles. This implies that the result (Theorem 1(c)) of H. Zoladek [Trans. Am. Math. Soc. 350, No. 4, 1681–1701 (1998; Zbl 0895.34026)] on the existence of algebraic limit cycles for Liénard systems is not correct. The third one characterizes all hyperelliptic limit cycles of Liénard systems with \(m=4\) and \(m+1<n<2m\). Moreover, the authors prove that there are Liénard systems which have \([m/2]-1\) algebraic limit cycles, where \([\cdot]\) denotes the integer part.