  A realistic generalization of the Markov--Dubins problem, which is concerned
with finding the shortest planar curve of constrained curvature joining two
points with prescribed tangents, is the requirement that the curve passes
through a number of prescribed intermediate points/nodes. We refer to this
generalization as the Markov--Dubins interpolation problem. We formulate this
interpolation problem as an optimal control problem and obtain results about
the structure of its solution using optimal control theory. The Markov--Dubins
interpolants consist of a concatenation of circular ($C$) and straight-line
($S$) segments. Abnormal interpolating curves are shown to exist and
characterized; however, if the interpolating curve contains a straight-line
segment then it cannot be abnormal. We derive results about the stationarity,
or criticality, of the feasible solutions of certain structure. In particular,
any feasible interpolant with arc types of $CSC$ in each stage is proved to be
stationary, i.e., critical. We propose a numerical method for computing
Markov--Dubins interpolating paths. We illustrate the theory and the numerical
approach by four qualitatively different examples.
