  We review in detail the Batalin-Vilkovisky formalism for Lagrangian field
theories and its mathematical foundations with an emphasis on higher algebraic
structures and classical field theories. In particular, we show how a field
theory gives rise to an $L_\infty$-algebra and how quasi-isomorphisms between
$L_\infty$-algebras correspond to classical equivalences of field theories. A
few experts may be familiar with parts of our discussion, however, the material
is presented from the perspective of a very general notion of a gauge theory.
We also make a number of new observations and present some new results. Most
importantly, we discuss in great detail higher (categorified) Chern-Simons
theories and give some useful shortcuts in usually rather involved
computations.
