Let A be a resolvent positive (linear) operator (i.e., \((\lambda -A)^{- 1}\) exists and is positive for \(\lambda >\lambda_ 0)\) on a Banach lattice E. Even though no norm condition on the resolvent is demanded, a theory is developed which - to a large extent - is analogous to the theory of positive \(C_ 0\)-semigroups. For example, if D(A) is dense or E is reflexive, then for every \(x\in D(A^ 2)\) there exists a unique classical solution of the abstract Cauchy problem \[ (ACP)\quad u(t)=Au(t)\quad (t\geq 0),\quad U(0)=x \] and u(t)\(\geq 0\) (t\(\geq 0)\) if \(x\geq 0\). Moreover, A is the generator of a so- called integrated semigroup; i.e. there exists S: [0,\(\infty)\to L(E)\) strongly continuous s.t. \((\lambda -A)^{-1}=\lambda \int^{\infty}_{0}e^{-\lambda t}S(t)dt\) \((\lambda >\lambda_ 0)\). The solution of (ACP) is given by \[ u(t)=S(t)Ax+x. \] A variety of examples is given.