Ageing phenomena and dynamical scaling behaviour have been observed in many out-of-equilibrium systems, but a general framework for the description of such systems is still missing. A first step in this direction is the theory of local scale-invariance (LSI), which attempts to identify generalised forms of spatio-temporal dynamical scaling. For systems with a dynamical exponent z = 2, it has already been known how to treat stochastic partial differential equations and the consequences have been verified in many explicit models. In this thesis a reformulation and extension of LSI for systems with z ? 2 is presented. We infer for the first time generalised Bargmann superselection and discuss extended dynamical symmetries of Langevin equations with z ? 2. We can establish a formalism for the calculation of non-equilibrium correlation -and response functions and the results are confirmed in several new model calculations. Secondly, the ageing behaviour in reaction-diffusion systems is investigated. Although the main features of ageing as seen in magnets are still valid, important differences in exponent relations are found. Explicitly, the contact process is studied through field-theoretical methods and two bosonic models are solved exactly. For the latter, we show how to extend LSI with z = 2 to nonlinear models. Thirdly, the ageing behaviour in semi-infinite magnetic systems close to the surface is considered. The results show that the general scaling picture known from infinite systems remains valid, but some ageing exponents and scaling functions differ from the bulk quantities.