The multinorm principle is a local-global principle for products of norm maps which generalizes the Hasse norm principle. Let L_1 and L_2 be finite separable extensions of a global field K. We say that an element of the multiplicative group of K is a local multinorm if it can be written as a product of norms of ideles from L_1 and L_2 and we say that such an element is a global multinorm if it can be written as a product of norms of field elements from L_1 and L_2. Then the pair of extensions L_1, L_2 satisfies the multinorm principle if every local multinorm is a global multinorm. Two basic problems are to determine which pairs of extensions satisfy the multinorm principle and to describe the obstruction to the multinorm principle which is defined as the group of local multinorms modulo the group of global multinorms. I will discuss what is known about each problem. In particular, I will sketch the computation of the obstruction for pairs of abelian extensions using class field theory, group cohomology, and the theory of Schur multipliers. I will also outline a purely cohomological approach to the multinorm problem which is based on the identification of the obstruction with the Tate-Shafarevich group of the associated multinorm torus.