We examine the theoretical and numerical global convergence properties of a certain "gradient free" stochastic approximation algorithm called the "simultaneous perturbation stochastic approximation (SPSA)" that has performed well in complex optimization problems. We establish two theorems on the global convergence of SPSA, the first involving the well-known method of injected noise. The second theorem establishes conditions under which "basic" SPSA without injected noise can achieve convergence in probability to a global optimum, a result with important practical benefits.