In 1957 Erd\"os studied trigonometric polynomials with all coefficients of absolute value 1 (called unimodular polynomials)and was led to the conjecture that the the maximum modulus of a unimodular polynomial of degree n is at least (1+c)\sqrt(n) for some positive absolute constant c. This conjecture was disproved by Littlewood in 1966 and, on the basis of numerical evidence, Littlewood conjectured that there are unimodular polynomials that deviate by o(\sqrt{n}) from its mean-square value \sqrt{n+1}. The existence of such polynomials, of any given degree, was proved by Kahane in 1980 using probabilistic methods, obtaining a remainder term of O(n^{1/2-1/17}\sqrt{log n}). In this lecture it will be given a new construction of these polynomials with the improved remainder term O(n^{1/2-1/9+epsilon}), first using probabilistic methods, and then with an explicit construction. The explicit construction makes use of Deligne's Riemann hypothesis for L-functions over varieties in positive characteristic associated to mixed exponential sums in arbitrarily many variables, as well as of sieves and recent results about gaps between squarefree numbers.