Given a directed weighted graph G = (V, E) undergoing vertex insertions and deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of G. In two breakthrough results, Italiano and Demetrescu [STOC'03] presented an algorithm that requires Õ(n²) amortized update time, and Thorup showed in [STOC'05] that worst-case update time Õ(n²⁺³/⁴) can be achieved. In this article, we make substantial progress on the problem. We present the following new results: • We present the first deterministic data structure that breaks the Õ(n²⁺³/⁴) worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time to Õ(n²⁺⁵/⁷) = Õ(n^2.71..) and to Õ(n²⁺³/⁵) = Õ(n^2.6) for unweighted graphs. • We present a simple deterministic algorithm with Õ(n²⁺³/⁴) worst-case update time (Õ(n²⁺²/³) for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time Õ(n²⁺²/³) (Õ(n²⁺¹/²) for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space Õ(n²). These are the first exact dynamic algorithms with truly-subcubic update time and space usage. This makes significant progress on an open question posed in multiple articles [COCOON'01, STOC'03, ICALP'04, Encyclopedia of Algorithms'08] and is critical to algorithms in practice [TALG'06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA'17].