Effects of the combination of nonlinearities and aperiodic order is studied in this thesis. The nonlinear systems considered are Josephson junction arrays (JJA) and Fermi-Pasta-Ulam (FPU) lattices. Both systems are discrete and one-dimensional. The traveling waves in a JJA are solitons, i.e., coherent structures which retain their shape and speed as they propagate. The properties of both these system is influenced by the ordering of their respective constituents. In the thesis some aperiodicaly orderd structures are described, including some typical effects on systems which are aper iodically orderd. Next, a description of Josephson junctions are given and also some physical features for system sconsisting of these junctions. A number of practical applications are given, Finally discrete breathers are described, including existence conditions and some possible applications.

Paper 1: Fluxon propagation in discrete Josepson junction arrays is examined. Sometimes a fluxon can be pinned in the array and we derive an effective potential which can explain why this happens, and predict at which positions a fluxon can get pinned.

Paper2; Discrete breathers are shown to exist in aperiodically ordered, diatomic FPU lattices. Localized modes survives for longer times in Fibonacci and Thue-Morse lattices compared to a periodic lattice. The conjecture is that the gaps in the linear phonon spectrum of the aperiodically orderd structures hinder resonances with the linear spectrum.leading to the longer life times.

Paper 3: Moving breathers in FPU lattices are examined. These have frequencies in the linear bands. Interaction with noise and imourity atoms are further considered.

Paper 4: Current-voltage (I-V) curves are calculated for Josephson junction arrays, which consist of two different types of junctions. A monoarray is compared to a periodic array and a Fibonacci array. There are marked differences in the I-V curves for these systems. New states with kink a structure superimposed on the rotating McCumber background are found in periodic array and in the Fibonacci array.