Title: Robust Approximate Zeros in Higher Dimensions. Abstract: Smale had introduced the notion of approximate zeros: If $f$ is an analytic function between two Banach spaces $E$, $F$ then a point $z_0$ is called an approximate zero of $f$ if there is a unique zero $z^*$ of $f$ such that the Newton iterates $z_i$ quadratically converge to $z^*$, i.e., $||z_i - z*^|| \le 2^{1-2^i}||z_0 - z^*||$. To computationally identify approximate zeros, Smale introduced the notion of point estimates. In particular, he showed that there is an easily computable function $\alpha(f,z)$ such that if $\alpha(f, z) < C$, for some universal global constant $C$, then $z$ is an approximate zero; one choice of $C$ is $0.03$. The problem with Smale's approach is that he assumes it is possible to compute the Newton iterates exactly. In practice, however, this is rarely the case so. We extend Smale's concept of approximate zeros to two computational models that account for errors in the computation: first, the weak model where the computations are done with a fixed precision; and second, the strong model where the computations are done with varying precision. For both models, we develop a notion of robust approximate zero and derive a corresponding robust point estimate. For the special case of a zero-dimensional system of integer polynomials, we bound the complexity of computing an absolute approximation to a root of the system using the strong model variant of Newton's method initiated from a robust approximate zero. The bound is expressed in terms of the condition number of the system and is a generalization of a well-known bound of Brent to higher dimensions. For implementation's sake, we also derive the explicit constants involved in the asymptotic bound on the precision required at each iteration of Newton's method in both the weak and the strong computational model.