We introduce a new family of codes, termed weighted Euclidean superimposed codes (WESCs). This family generalizes the class of Euclidean superimposed codes, used in multiuser identification systems. WESCs allow for deterministic discrimination of bounded, integer-valued linear combinations of real-valued codewords, and can therefore also be seen as a specialization of compressed sensing schemes. We first present lower and upper bounds on the largest size a WESC, and show how to use classical coding-theoretic and new compressed sensing analytical tools to devise low-complexity decoding algorithms for these codes. Then we study sparse WESCs, in which the number of nonzero coordinates of codewords are significantly smaller than their length. Our results include a lower bound on the largest rate of sparse WESCs and a deterministic construction based on DeVore's method.