Abstract: A Hadamard matrix H is an n-by-n matrix with -1,+1 elements such that H H^t = n I_n, where t denotes transposition and I_n denotes the n-by-n identity matrix. Hadamard matrices arise in Statistics, Combinatorics, Coding Theory, Cryptography, Telecommunications and other areas. A necessary condition for the existence of a Hadamard matrix is that n = 1, 2, or a multiple of 4. The sufficiency of this statement, i.e. that for every n a multiple of 4 there exists a Hadamard matrix of order n is the celebrated Hadamard conjecture. The smallest order n for which a Hadamard matrix is still unknown, is n = 668. A number of approaches to tackle the Hadamard conjecture have been suggested in the last 120 years. We will discuss some old and new approaches to the Hadamard conjecture. These approaches are based on sequences with zero periodic/non periodic autocorrelation functions, Computational Algebra and Coding Theory.