High-Reynolds-number unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite moving cylinder with time-dependent axial velocity and with uniform transpiration, , is investigated. The impinging free stream is steady and with a strain rate . An exact solution of the Navier-Stokes equations and energy equation is derived in this problem. The general self-similar solution is obtained when the axial velocity of the cylinder and its wall temperature or its wall heat flux vary as certain functions. These solutions are presented for special cases when the time-dependent axial velocity of the cylinder and its wall temperature or heat flux are certain functions of time. All the solutions above are presented for Reynolds numbers, , ranging from 0.1 to 1000 and different values of dimensionless transpiration rate, , where is cylinder radius and is kinematic viscosity of the fluid. Shear stresses corresponding to all the cases increase with the Reynolds number and decrease with the increase of suction rate. Using an inner-outer expansion, uniformly valid solutions of the equations in the case of high Reynolds numbers are obtained.