LCL filters have been widely used for grid-connected inverters. However, the problem that how time delay affects the stability of digitally controlled grid-connected inverters with LCL filters has not been fully studied. In this paper, a systematic study is carried out on the relationship between the time delay and stability of single-loop controlled grid-connected inverters that employ inverter current feedback (ICF) or grid current feedback (GCF). The ranges of time delay for system stability are analyzed and deduced in the continuous s-domain and discrete z-domain. It is shown that in the optimal range, the existence of time delay weakens the stability of the ICF loop, whereas a proper time delay is required for the GCF loop. The present work explains, for the first time, why different conclusions on the stability of ICF loop and GCF loop have been drawn in previous studies. To improve system stability, a linear predictor-based time delay reduction method is proposed for ICF, while a time delay addition method is used for GCF. A controller design method is then presented that guarantees adequate stability margins. The delay-dependent stability study is verified by simulation and experiment.