Web-scale search systems learn an encoder to embed a given query which is then hooked into an approximate nearest neighbor search (ANNS) pipeline to retrieve similar data points. To accurately capture tail queries and data points, learned representations typically are _rigid, high-dimensional_ vectors that are generally used as-is in the entire ANNS pipeline and can lead to computationally expensive retrieval. In this paper, we argue that instead of rigid representations, different stages of ANNS can leverage _adaptive representations_ of varying capacities to achieve significantly better accuracy-compute trade-offs, i.e., stages of ANNS that can get away with more approximate computation should use a lower-capacity representation of the same data point. To this end, we introduce AdANNS, a novel ANNS design framework that explicitly leverages the flexibility of Matryoshka Representations. We demonstrate state-of-the-art accuracy-compute trade-offs using novel AdANNS-based key ANNS building blocks like search data structures (AdANNS-IVF) and quantization (AdANNS-OPQ). For example on ImageNet retrieval, AdANNS-IVF is up to $\mathbf{1.5}$% more accurate than the rigid representations-based IVF at the same compute budget; and matches accuracy while being up to $\mathbf{90}\times$ faster in _wall-clock time_. For Natural Questions, $32$-byte AdANNS-OPQ matches the accuracy of the $64$-byte OPQ baseline constructed using rigid representations -- _same accuracy at half the cost!_ We further show that the gains from AdANNS translate to modern-day composite ANNS indices that combine search structures and quantization. Finally, we demonstrate that AdANNS can enable inference-time adaptivity for compute-aware search on ANNS indices built non-adaptively on matryoshka representations. Code is open-sourced at https://github.com/RAIVNLab/AdANNS.