This thesis studies singularly perturbed Volterra integral equations of the form eu(t)=/(t,e)+fg(t,s,11(5)) ds, 0<iJo_<r, where e>0 is a small parameter The function f(t,e) is defined for 0<t<T anil g(t,s,u) for 0<s<t<T There are many existence and uniqueness results known that ensure ¡that a unique continuous solution u(t, e) exists for all small e>0. The aim is to find asymptotic approximations l to these solutions. This work is restricted to problems where there is an imtial-layer, various hypotheses are placed on g(tts,u) to exclude other behaviour. A major part of this work is that formal solutions of the nonlinear problem are determined and rigorously proved to be asymptotic approximations to the exact solutions Formal approximate solutions: £Mi,e) NEn-0 Enu(t,E,),=un(t,e)=0(1) as e->0, are obtained using the additive decomposition method Algorithms which improve the method used m Angell and Olmstead (1987), are presented for obtaining these solutions Assuming a stability condition in the boundary layer, it shown that there is a constant Cjv such ,that \u(t,e)-Ujsf{t,e)\ <cN eN+1 as e 0, uniformly for t € [0,T], thus establishing that £/jv(i,£) is an asymptotic solution SIkinner (1995) has proved similar results, but almost all the theorems here were discovered before Skinner’s work was found and are largely independent of it Lange and Smith (1988) prove results for the case g(t, syu) = k(tys)u, where k(t, s) is continuous and satisfies a stability condition in the boundary layer These results are carefully developed here and similar results for linear integrodifferential equations The problem of extending these to the class of weakly singular equations with g(t,s,u)= k(t,s)/(t-s)BU-0</3<1 is discussed An interesting aspect of this problem and others for which the boundary 1 layer stability condition fails, is that the solutions decay algebraically rather exponentially within ¡the boundary layer.