We present a new extension of the generalized caching/paging problem that allows the adversary to arbitrarily change the cost or weight of the currently requested page. We present modifications of previous algorithms for generalized caching to handle varying page weights and page costs. In particular, a deterministic algorithm based on [5, 9] for an (h, k)-competitive algorithm with competitive ratio k/(k − h + 1) is presented. In addition, a randomized algorithm based on [1, 2] with competitive ratio O(log k) is presented. We present three applications that can be supported via reductions to generalized caching with varying page weights and page costs. These applications are: (1) support of subsets of pages that must be simultaneously present in the cache before entry to a critical section (i.e., working sets), (2) change of page size due to compression and decompression, (3) variable cache size (i.e., elastic caches).