Form a random k-SAT formula on n variables by selecting uniformly and independently m = rn clauses out of all 2^k (_k^n ) possible k-clauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r_k such that as n tends to infinity, the probability that the formula is satisfiable tends to 1 if r < r_k and to 0 if r > r_k. It has long been known that 2 k/k < r_k > 2^k. We prove that r_k > 2^{k - 1} ln 2 - d_k, where d_k → (1 + ln 2)/2. Our proof also allows a blurry glimpse of the "geometry" of the set of satisfying truth assignments.