Random Bernoulli matrices (the entries are i.i.d Bernoulli random variables, taking value 1 and -1 with probability half) are of interest in several areas: combinatorics, mathematical physics, theoretical computer science, to mention a few. On the other hand, very little has been proved about this model. In this talk, I am going to discuss several basic and rather notorious questions and mention few recent results, obtained with Terry Tao. \vskip .1in \noindent Some detailed proofs will be presented in a subsequent seminar.