Recent discovery of superconductivity (SC) in the novel quasi-1D system A2Cr3As3 (A=K, Rb and Cs with TC ≈ 6.1 K, 4.8 K and 2.2 K, respectively) has attracted considerable interest. The unconventional SC has been revealed by both experimental and theoretical investigations. As theoretical calculations have pointed out that K2Cr3As3 is near magnetic order and SC is relevant to strong spin fluctuations, we propose to carry out muon spin rotation and relaxation (µSR) measurements on A2Cr3As3 (A=K, Rb and Cs) using the MUSR spectrometer to probe the superconducting state, nature of the gap, pairing mechanism and order parameters. The proposed µSR study will determine the superconducting ground state property and pairing mechanism and give the key information on the nature of the SC that will allow to test the theoretical predictions.