  Let $L\subset X$ be a not necessarily orientable relatively $Pin$ Lagrangian
submanifold in a symplectic manifold $X$. Evaluation maps of moduli spaces of
$J$-holomorphic disks with boundary in $L$ may not be relatively orientable. To
deal with this problem, we introduce the notion of an orientor. Interactions
between the natural operations on orientors are governed by orientor calculus.
Orientor calculus gives rise to a family of orientors on moduli spaces of
$J$-holomorphic stable disk maps with boundary in $L$ that satisfy natural
relations. In a sequel, we use these orientors and relations to construct the
Fukaya $A_\infty$ algebra of $L.$
