In this paper, we study binary signature codes for the weighted binary adder channel (WbAC) and collusion-resistant multimedia fingerprinting. Let A(n, t) denote the maximum size of a t-signature code of length n, and A(n, w, t) denote the maximum size of a t-signature code of length n and constant-weight w. First, we derive asymptotic and general upper bounds on A(n, t) by relating signature codes to Bt codes and bipartite graphs with large girth respectively, and also show the upper bounds are tight for certain cases. Second, we determine the exact values of A(n, 2, 2) and A(n, 3, 2) for infinitely many n by connecting signature codes with C4-free graphs and union-free families, respectively. Third, we provide two explicit constructions for t-signature codes which have efficient decoding algorithms and applications to two-level signature codes. Furthermore, we show from a geometric viewpoint that there does not exist any binary code with complete traceability for noisy WbAC and multimedia fingerprinting. A new type of signature codes with a weaker requirement than complete traceability is introduced for the noisy scenario.