Resolvent positive operators on an ordered Banach space (with generating and normal positive cone) are by definition linear (possibly unbounded) operators whose resolvent exists and is positive on a right half‐line. Even though these operators are defined by a simple (purely algebraic) condition, analogues of the basic results of the theory ofC0‐semigroups can be proved for them. In fact, ifAis resolvent positive and has a dense domain, then the Cauchy problem associated withAhas a unique solution for every initial value in the domain ofA2, and the solution is positive if the initial value is positive. Also the converse is true (if we assume thatAhas a non‐empty resolvent set andD(A2)∩E+is dense inE+). Moreover, every positive resolvent is a Laplace–‐Stieltjes transform of a so‐called integrated semigroup; and conversely every such (increasing, non‐degenerate) integrated semigroup defines a unique resolvent positive operator.