The main object in this paper is the sequence of polynomials P_n(z) that have Stern numbers as their coefficients; that is, the terms of Stern's diatomic sequence. We derive certain basic properties of these polynomials, investigate the distribution of their real and complex zeros, and prove some results concerning factorizations and resultants. We also consider the (0,1)-polynomials obtained from P_n(z) by taking their coefficients modulo 2. In spite of its simple form, the polynomial sequence is shown to possess some interesting algebraic and analytic properties. Finally, we discuss combinatorial interpretations of the polynomials P_n(z) and indicate ways of generalizing them.