Most current Model Reference Adaptive Control (MRAC) methods rely on parametric adaptive elements, in which the number of parameters of the adaptive element are fixed a-priori, often through expert judgment. Examples of such adaptive elements are the commonly used Radial Basis Function Networks (RBFNs) with pre-allocated centers allocated based on the expected operating domain. A severe limitation of such RBFN MRAC methods is that if the system operates outside of the expected operating domain, such adaptive elements can become non-effective, thus rendering the adaptive controller only semi-global in nature. This paper treats system uncertainties as distributions over functions and proposes Gaussian Process based adaptive elements. We show that these Bayesian nonparametric adaptive elements guarantee good closed loop performance with minimal prior domain knowledge of the uncertainty through stochastic stability arguments. Online implementable GP inference method are evaluated in simulations and compared with RBFN adaptive controllers with pre-allocated centers. The results indicate that GP-MRAC overcomes the limitations of MRAC employing RBFN with fixed parameters.