Growth of quantum-confined semiconductor structures is a complicated process that may lead to imperfect and complex shapes as well as geometrical nonuniformities when comparing a large number of intended identical structures. On the other hand, the possibility of tuning the shape and size of nanostructures allows for extra optimization degrees when considering electronic and optical properties in various applications. This calls for a better understanding of size and shape effects. In the present work, we express the one-band Schrödinger equation in curved coordinates convenient for determining eigenstates of curved quantum-wire and quantum-dash structures with large aspect ratios. Firstly, we use this formulation to solve the problem of single-electron and single-hole states in curved nanowires. Secondly, exciton states for the curved quantum-wire Hamiltonian problem are found by expanding exciton eigenstates on a product of single-particle eigenstates. A simple result is found for the Coulomb matrix elements of an arbitrarily curved structure as long as the radius-of-curvature is much larger than the cross-sectional dimensions. We use this general result to compute the groundstate exciton binding energy of a bent nanowire as a function of the bending radius-of-curvature. It is demonstrated that the groundstate exciton binding energy increases by 40 meV as the radius-of-curvature changes from 20 to 2 nm while keeping the total length (and volume) of the nanowire constant.