In this paper, we introduce a new family of Schur functions that depend on two sets of variables and two doubly infinite sequences of parameters. These functions generalize and unify various existing Schur functions, including classical Schur functions, factorial Schur functions, supersymmetric Schur functions, Frobenius-Schur functions, factorial supersymmetric Schur functions, and dual Schur functions. We prove that the new family of functions satisfies several well-known properties, such as the combinatorial description, Jacobi-Trudi identity, Nägelsbach-Kostka formula, Giambelli formula, Ribbon formula, Weyl formula, Berele-Regev factorization, and Cauchy identity.Our approach is based on the integrable six vertex model with free fermionic weights. We show that these weights satisfy the refined Yang-Baxter equation, which results in supersymmetry for the Schur functions. Furthermore, we derive refined operator relations for the row transfer operators and use them to find partition functions with various boundary conditions. Our results provide new proofs for known results as well as new identities for the Schur functions. Contents 18 3. Partitions, Maya diagrams, and ribbons 22 4. Free fermionic Schur functions 23 4.1. Further properties 28 References 29