  In this text we develop the formalism of products and powers of linear codes
under componentwise multiplication. As an expanded version of the author's talk
at AGCT-14, focus is put mostly on basic properties and descriptive statements
that could otherwise probably not fit in a regular research paper. On the other
hand, more advanced results and applications are only quickly mentioned with
references to the literature. We also point out a few open problems.
  Our presentation alternates between two points of view, which the theory
intertwines in an essential way: that of combinatorial coding, and that of
algebraic geometry.
  In appendices that can be read independently, we investigate topics in
multilinear algebra over finite fields, notably we establish a criterion for a
symmetric multilinear map to admit a symmetric algorithm, or equivalently, for
a symmetric tensor to decompose as a sum of elementary symmetric tensors.
