This paper proposes a spatially common sparsity based adaptive channel estimation and feedback scheme for frequency division duplex based massive multi-input multi-output (MIMO) systems, which adapts training overhead and pilot design to reliably estimate and feed back the downlink channel state information (CSI) with significantly reduced overhead. Specifically, a non-orthogonal downlink pilot design is first proposed, which is very different from standard orthogonal pilots. By exploiting the spatially common sparsity of massive MIMO channels, a compressive sensing (CS) based adaptive CSI acquisition scheme is proposed, where the consumed time slot overhead only adaptively depends on the sparsity level of the channels. Additionally, a distributed sparsity adaptive matching pursuit algorithm is proposed to jointly estimate the channels of multiple subcarriers. Furthermore, by exploiting the temporal channel correlation, a closed-loop channel tracking scheme is provided, which adaptively designs the non-orthogonal pilot according to the previous channel estimation to achieve an enhanced CSI acquisition. Finally, we generalize the results of the multiple-measurement-vectors case in CS and derive the Cramer-Rao lower bound of the proposed scheme, which enlightens us to design the non-orthogonal pilot signals for the improved performance. Simulation results demonstrate that the proposed scheme outperforms its counterparts, and it is capable of approaching the performance bound.