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In this paper, we propose AdaBB, an adaptive gradient method based on the Barzilai-Borwein stepsize. The algorithm is line-search-free and parameter-free, and it essentially provides a convergent variant of the Barzilai-Borwein method for general convex optimization problems. We analyze the ergodic convergence of the objective function value and the convergence of the iterates for solving general convex optimization problems. Compared with existing works along this line of research, our algorithm gives the best lower bounds on the stepsize and the average of the stepsizes. Furthermore, we present extensions of the proposed algorithm for solving locally strongly convex and composite convex optimization problems where the objective function is the sum of a smooth function and a nonsmooth function. In the case of local strong convexity, we achieve linear convergence. Our numerical results also demonstrate very promising potential of the proposed algorithms on some representative examples. Funding: S. Ma is supported by the National Science Foundation [Grants DMS-2243650, CCF-2308597, CCF-2311275, and ECCS-2326591] and a startup fund from Rice University. J. Yang is supported by the National Natural Science Foundation of China [Grants 12431011 and 12371301] and the Natural Science Foundation for Distinguished Young Scholars of Gansu Province [Grant 22JR5RA223].