Spin-based computing schemes could enable new functionalities beyond those of charge-based approaches(1-6). Examples include nanomagnetic logic, where information can be processed using dipole coupled nanomagnets(7-14), as demonstrated by multi-bit computing gates(8,13). One fundamental benefit of using magnets is the possibility of a significant reduction in the energy per bit compared with conventional transistors(1,8,14,15). However, so far, practical implementations of nanomagnetic logic have been limited by the necessity to apply a magnetic field for clocking(8,11). Although the energy associated with magnetic switching itself could be very small, the energy necessary to generate the magnetic field renders the overall logic scheme uncompetitive when compared with complementary metaloxide-semiconductor (CMOS) counterparts. Here, we demonstrate a nanomagnetic logic scheme at room temperature where the necessity for using a magnetic field clock can be completely removed by using spin-orbit torques(16-22). We construct a chain of three perpendicularly polarized CoFeB nanomagnets on top of a tantalum wire and show that an unpolarized current flowing through the wire can 'clock' the perpendicular magnetization to a metastable state. An input magnet can then drive the nanomagnetic chain deterministically to one of two dipole-coupled states, '2 up 1 down' or '2 down 1 up', depending on its own polarization. Thus, information can flow along the chain, dictated by the input magnet and clocked solely by a charge current in tantalum, without any magnetic field. A three to four order of magnitude reduction in energy dissipation is expected for our scheme when compared with state-of-the-art nanomagnetic logic.