Numerous problems in science and technology can be described with the help of partial differential equations. This thesis is dedicated to the numerical treatment of these equations using adaptive quarklet methods. By a quarklet we understand the product of a wavelet and a piecewise polynomial, consequently we call a polynomially enriched wavelet basis quarklet frame. These function systems can initially be constructed both on the real line and the unit interval and, having done this, be generalised to higher dimensions by using tensor product techniques and domain decompositions. Quarklet systems are stable in Besov and Sobolev spaces, furthermore they fulfil certain compressibility properties and hence are convenient to be utilised in generic frame methods for the treatment of operator equations. Adaptive quarklet methods represent hp-variants of wavelet methods, therefore there is strong hope that they converge quite fast.