Abstract: Planar spiral waves are rotating waves that rotate with constant angular velocity and behave like Archimedean spirals in their far field. Such spirals arise in many experiments as well as in reaction-diffusion models. In this thesis, the persistence of planar spiral waves is investigated upon excising a small hole from the domain near the core region of the spiral. Before treating 2D patterns, we investigate the persistence of 1D pulses upon truncating the real line to large but bounded intervals, supplemented by boundary conditions at their end points. Under appropriate transversality assumptions on the boundary conditions, the persistence of pulses is established and their stability with respect to the reaction-diffusion system on the bounded interval is determined. It turns out that the stability properties of the truncated pulses depends on the choice of boundary conditions. These results are then applied to the front of the Nagumo equation and the fast pulse of the FitzHugh-Nagumo equation. In the second part of the thesis, we analyse the persistence of 2D spiral waves by posing the elliptic partial differential equation as an ill-posed dynamical system in which the radius serves as the time-like variable. In this setting, the approach via Lyapunov-Schmidt reduction and Lin's method utilized in the one-dimensional case carries over to systems posed on the plane. Upon establishing suitable a-priori estimates for the dynamics on the center eigenspace, we prove the persistence of core-region spirals that satisfy the boundary conditions, and afterwards match with far-field spirals to obtain a unique planar spiral that obeys the boundary conditions.