The aim of this book is to outline an elementary theory of linear functional on convex cones, but convex cones are here taken in a slightly more general way than usual, they need not be imbedded in a vector space. In consequence, we do not have a general cancellation law for the addition. Typical examples for the cones we have in mind are R = R ? {- <*>} or the upper semicontinuous R- valued functions on some topological space. Accordingly, linear functionals on such cones are allowed to attain values in R instead of R . This generality has advantages with respect to extensions of linear functionals.