We investigate the numbers of complex zeros of Littlewood polynomials $p\left( z \right)$ (polynomials
with coefficients {−1, 1}) inside or on the unit circle $\left| z \right|\,=\,1$, denoted by $N\left( p \right)$ and $U\left( p \right)$, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in
the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain
explicit formulas for $N\left( p \right)$, $U\left( p \right)$ for polynomials $p\left( z \right)$ of these types. We show that if $n\,+\,1$ is a prime number, then for each integer $k,\,0\,⩽\,k\,⩽\,n-1$, there exists a Littlewood polynomial $p\left( z \right)$ of degree $n$ with $N\left( p \right)\,=\,k$ and $U\left( p \right)\,=\,0$. Furthermore, we describe some cases where the ratios $N\left( p \right)/n$ and $U\left( p \right)/n$ have limits as $n\,\to \,\infty $ and find the corresponding limit values.