The algebraic theory of corner subrings introduced by Lam (as an abstraction of the properties of Peirce corners eRe of a ring R associated with an idempotent e in R) is investigated here in the context of Banach and C*-algebras. We propose a general algebraic approach which includes the notion of ranges of (completely) contractive conditional expectations on C*-algebras and on ternary rings of operators, and we investigate when topological properties are consequences of the algebraic assumptions. For commutative C*-algebras we show that dense corners cannot be proper and that self-adjoint corners must be closed and always have closed complements (and may also have non-closed complements). For C*-algebras we show that Peirce corners and some more general corners are similar to self-adjoint corners. We show uniqueness of complements for certain classes of corners in general C*-algebras, and establish that a primitive C*-algebra must be prime if it has a prime Peirce corner. Further we consider corners in ternary rings of operators (TROs) and characterise corners of Hilbertian TROs as closed subspaces.