This paper considers massive access in massive multiple-input multiple-output (MIMO) systems and proposes an adaptive active user detection and channel estimation scheme based on compressive sensing. By exploiting the sporadic traffic of massive connected user equipments and the virtual angular domain sparsity of massive MIMO channels, the proposed scheme can support massive access with dramatically reduced access latency. Specifically, we design non-orthogonal pseudo-random pilots for uplink broadband massive access, and formulate the active user detection and channel estimation as a generalized multiple measurement vector compressive sensing problem. Furthermore, by leveraging the structured sparsity of the uplink channel matrix, we propose an efficient generalized multiple measurement vector approximate message passing (GMMV-AMP) algorithm to realize joint active user detection and channel estimation based on a spatial domain or an angular domain channel model. To jointly exploit the channel sparsity present in both the spatial and the angular domains for enhanced performance, a Turbo-GMMV-AMP algorithm is developed for detecting the active users and estimating their channels in an alternating manner. Finally, an adaptive access scheme is proposed, which adapts the access latency to guarantee reliable massive access for practical systems with unknown channel sparsity level. Additionally, the state evolution of the proposed GMMV-AMP algorithm is derived to predict its performance. Simulation results demonstrate the superiority of the proposed active user detection and channel estimation schemes compared to several baseline schemes.