Delay systems used to model retarded actions are relevant in many fields such as optics, mechanical machining, biology or physiology. A frequently encountered situation is that the length of the delay time changes with time. In this study, we use a simple map system to investigate the influence of the fluctuating delay time on the system dynamics. For simplicity, we start from a case with the delay time taking only the value of zero or one discrete time steps, where the system dynamics reduces to one- and two-dimensional map, respectively. We study two situations, periodic or random variation of the delay. Rich dynamics including coexisting multiple attractors, strange nonchaotic attractors and on-off intermittency are observed. For a special case we can show analytically the existence of a dense set of singularities of the Lyapunov exponent. Finally we present results for higher dimensional generalizations to show the relevance of our findings to more general situations.