Unsteady creeping flow of a liquid that wets a solid is considered. The liquid/gas interface at a distance from the wetting line has a shape of meniscus. Asymptotic methods are involved to evaluate the influence of the unsteady motion of the meniscus contact line on viscous flow and meniscus contact angle. An asymptotic relation for the dynamic contact angle in the case of unsteady wetting is derived. The meniscus contact angle is shown to depend also on both velocity and acceleration in the wetting process. The general solution applies to a wide range of problems. For example, it describes meniscus motion in a channel with a variable flow rate of a medium affected by gravity. Consideration is given to a complete wetting and to a surface covered with a thin macroscopic film. General asymptotic relations are proposed for describing the process of narrowing of the contact angle of an arbitrary meniscus during transition to equilibrium. The characteristic time of relaxation of the dynamic contact angle is established; critical conditions of dynamic contact angle "disappearance" are outlined.