This book aims to develop a general integration theory for stochastic processes with stationary increments and spectral density. This class of motions particularly allows the simultaneous study of long-range dependence and intermittency effects and includes the most relevant random processes used in modern stochastic analysis. So for instance the Wiener process, the fractional Brownian motion, the fractional Riesz-Bessel motion but also Poisson and Levy processes. The so obtained knowledge on generalised stochastic integration will be used to achieve regularity results and is applied to parabolic Volterra problems with random noise as well as to the problem of anomalous diffusion with stochastic disturbance along the boundary.