Functions satisfying the triangle inequality $d(x,y)\leq d(x,z)+d(z,y)$, but lacking symmetry $d(x,y)=d(y,x)$, have been studied in various fields like fuzzy logic, category theory, and non-symmetric topology. They also arise in C*-algebras as the appropriate continuous/fuzzy versions of well-studied order relations. In our talk, we outline their basic theory, showing how they unify aspects of metric and order theory, and give some examples and applications in operator algebras.