  We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and
Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the
open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| < 1\}$. For every pair
$(k, n) \in \mathbb{N}^2$, where $n \geq 7$ and $k \in [3, n-3]$, we prove that
it is possible to find a $\{0, 1\}$--polynomial $f(z)$ of degree $\text{deg
}{f}=n$ with non--zero constant term $f(0) \ne 0$, such that $N(f)=k$ and $f(z)
\ne 0$ on the unit circle $\partial\mathcal{D}$. On the way to this goal, we
answer a question of D.~W.~Boyd from 1986 on the smallest degree Newman
polynomial that satisfies $|f(z)| > 2$ on the unit circle $\partial
\mathcal{D}$. This polynomial is of degree $38$ and we use this special
polynomial in our constructions. We also identify (without a proof) all
exceptional $(k, n)$ with $k \in \{1, 2, 3, n-3, n-2, n-1\}$, for which no such
$\{0, 1\}$--polynomial of degree $n$ exists: such pairs are related to regular
(real and complex) Pisot numbers.
  Similar, but less complete results for $\{-1, 1\}$ polynomials are
established. We also look at the products of spaced Newman polynomials and
consider the rotated large Littlewood polynomials. Lastly, based on our data,
we formulate a natural conjecture about the statistical distribution of $N(f)$
in the set of Newman and Littlewood polynomials.
