Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (v, k, λ)-designs. When v = 4t-1, k = 2t -1 and λ = t-1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, v is either a prime number, a product of twin primes, or one less than a power of 2. Except when v = 2k-1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when v = p(p+2) where both p and p+2 are prime); or sextic residues (when v is a prime of the form 4a2 + 27). However, when v = 2k-1, many constructions are now known, including m-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2k-1 is prime), GMW sequences and their generalizations (when k is composite), certain term-by-term sums of three and of five m-sequences and more general sums of trace terms, several constructions based on hyper-ovals in finite geometries (found by Segre, by Glynn, and by Maschietti), and the result of performing the Welch-Gong transformation on some of the foregoing.