Variational Quantum Algorithms (VQAs) are viewed as amongst the best hope for near-term quantum advantage. A natural question is whether noise places fundamental limitations on VQA performance. In the first part of this talk, we show that noise can severely limit the trainability of VQAs by exponentially flattening the optimization landscape and suppressing the magnitudes of cost gradients. Specifically, for the class of local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n. In the second part of this talk, we consider whether error mitigation (EM) techniques can remedy these effects and improve trainability. We find that for a broad class of EM strategies, these scaling effects due to noise cannot be resolved without committing exponential resources elsewhere. Moreover, EM can even itself further impair trainability.