The accuracy of the finite dfference technique in solving frictionless and frictional advancing contact problems is investigated by solving the problem of a rigid punch on an elastic half- space subjected to normal loading. Stick and slip conditions between the elastic and the rigid materials are added to an existing numerical algorithm which was previously used for solving frictionless and frictional stationary and receding contact problems. The numerical additions are first tested by applying them in the solution of receding and stationary contact problems and comparing them to known solutions, The receding contact problem is that of an elastic slab on a rigid half-plane; the stationary contact problem is that of aflat rigid punch on an elastic half-space, In both cases the influence offriction is examined. The results are compared to those of other investigations with very good agreement observed. Once more it is verfied that for both receding and stationary contact, load steps are not required for obtaining a solution f the loads are applied monotonically, whether or not there is friction. Next, an advancing contact problem of a round rigid punch on an elastic half-space subjected to normal loading, with and without the influence offriction is investigated. The results for frictionless advancing contact, which are obtained without load steps, are compared to analytical results, namely the Hertz problem; excellent agreement is observed When friction is present, load steps and iterations for determining the contact area within each load step, are required. Hence, the existing code, in which only iterations to determine the contact zone were employed, was mod(fiedto include load steps, together with the above mentioned iterations for each load step. The effect offriction on the stress distribution and contact length is studied. It isJound that when stick conditions appear in the contact zone, an increase in the friction coefficient results in an increase in the stick zone size within the contact zone. These results agree well with semi- analytical results of another investigation, illustrating the accuracy and capabilities of the finite dfference technique for advancing contact.