In this paper, we study the problem of massive connections over limited radio spectrum for the cellular Internet-of-Things (IoT) in the fifth-generation (5G) wireless network. To address the challenging issues associated with channel state information (CSI) acquisition and beam design in the context of massive connections, we propose a new non-orthogonal beamspace multiple access framework. In particular, the user equipments (UEs) are non-orthogonal not only in the temporal-frequency domain, but also in the beam domain. We analyze the performance of the proposed non-orthogonal beamspace multiple access scheme, and derive an upper bound on the weighted sum rate in terms of channel conditions and system parameters. For further improving the performance, we propose three non-orthogonal transmit beam construction algorithms with different beamspace resolutions. Finally, extensive simulation results show that substantial performance gain can be obtained by the proposed non-orthogonal beamspace multiple access scheme over the baseline ones.