  We study the problem of minimizing the supremum norm, on a segment of the
real line or on a compact set in the plane, by polynomials with integer
coefficients. The extremal polynomials are naturally called integer Chebyshev
polynomials. Their factors, zero distribution and asymptotics are the main
subjects of this paper. In particular, we show that the integer Chebyshev
polynomials for any infinite subset of the real line must have infinitely many
distinct factors, which answers a question of Borwein and Erd\'{e}lyi.
Furthermore, it is proved that the accumulation set for their zeros must be of
positive capacity in this case.
  We also find the first nontrivial examples of explicit integer Chebyshev
constants for certain classes of lemniscates. Since it is rarely possible to
obtain an exact value of integer Chebyshev constant, good estimates are of
special importance.
  Introducing the methods of weighted potential theory, we generalize and
improve the Hilbert-Fekete upper bound for integer Chebyshev constant. These
methods also give bounds for the multiplicities of factors of integer Chebyshev
polynomials, and lower bounds for integer Chebyshev constant. Moreover, all the
mentioned bounds can be found numerically, by using various extremal point
techniques, such as weighted Leja points algorithm. Applying our results in the
classical case of the segment $[0,1]$, we improve the known bounds for the
integer Chebyshev constant and the multiplicities of factors of the integer
Chebyshev polynomials.
