In modern distributed systems, failures are the norm rather than the exception. In many cases, these failures are not benign. Settings such as the Internet might incur malicious (also called Byzantine or arbitrary) behavior and asynchrony. As a result, and perhaps not surprisingly, research on asynchronous Byzantine fault-tolerant (BFT) distributed systems is flourishing. Tolerating arbitrary behavior and asynchrony calls for very sophisticated algorithms. This is in particular the case with BFT solutions that aim to provide properties such as: (a) optimal resilience, i.e., tolerating as many Byzantine failures as possible and (b) optimal performance with respect to some relevant complexity metric. Most BFT algorithms are built from scratch or by modifying existing solutions in a non-modular manner, which often renders these algorithms difficult to understand and, consequently, impedes their wider adoption. We attribute this complexity to the lack of sufficient number of adequate abstractions for asynchronous BFT distributed computing. The motivation of this thesis is to propose reusable abstractions for devising asynchronous BFT distributed algorithms that are optimally resilient and/or have optimal complexity, with strong focus on one of the most important complexity metrics — time complexity (or latency). The abstractions proposed in this thesis are devised with three fundamental distributed applications in mind: (a) read/write storage (also called register), (b) consensus and (c) state machine replication (SMR). We demonstrate how to use our abstractions in these applications to devise asynchronous BFT algorithms that feature the best complexity among all algorithms we know of, in addition to optimal resilience. First, we introduce the notion of a refined quorum system (RQS) of some set S as a set of three classes of subsets (quorums) of S: first class quorums are also second class quorums, themselves being also third class quorums. First class quorums have large intersections with all other quorums, second class quorums typically have smaller intersections with those of the third class, the latter simply correspond to traditional quorums. The refined quorum system abstraction helps design algorithms that tolerate contention (process concurrency), arbitrarily long periods of asynchrony and the largest possible number of failures, but perform fast if few failures occur, the system is synchronous and there is no contention, i.e., under conditions that are assumed to be frequent in practice. In other words, RQS helps combine optimal resilience and optimal best-case time complexity. Intuitively, under uncontended and synchronous conditions, a distributed object implementation would expedite an operation if a quorum of the first class is accessed, then degrade gracefully depending on whether a quorum of the second or the third class is accessed. Our notion of RQS is devised assuming a general adversary structure, and this basically allows algorithms relying on RQS to relax the assumption of independent process failures. We illustrate the power of refined quorums by introducing two new optimal BFT atomic object implementations: an atomic storage and consensus algorithm. Our second abstraction is a novel timestamping mechanism called high resolution timestamps (HRts), which can be seen as a variation of a matrix clocks. Roughly speaking, a high resolution timestamp contains a matrix of local timestamps of (a subset of) processes as seen by (a subset of) other processes. Complementary to RQS, HRts simplify the design of BFT distributed algorithms that combine optimal resilience and worst-case time complexity. We apply high-resolution timestamps to design read/write storage algorithms in which HRts are used to detect and filter out Byzantine processes, which paves the path to the first BFT storage algorithms that combine optimal resilience with optimal worst-case time complexity. Finally, we introduce ABsTRACT (Abortable Byzantine faulT-toleRant stAte maChine replicaTion), a generic abstraction that simplifies the notoriously difficult task of developing BFT state machine replication algorithms. ABsTRACT resembles BFT-SMR and it can be used to make any shared service Byzantine fault-tolerant, with one exception: it may sometimes abort a client request. The non-triviality condition under which ABsTRACT cannot abort is a generic parameter. We view a BFT-SMR algorithm as a composition of instances of ABsTRACT, each instance developed and analyzed independently. To illustrate our approach, we describe two new optimally resilient BFT algorithms. The first, that makes use of our refined quorums, has the lowest time complexity among all BFT-SMR algorithms we know of, in synchronous periods that are free from contention and failures. The second algorithm has the highest peak throughput in failure-free and synchronous periods; this algorithm argues for general applicability of ABsTRACT in developing BFT shared services that feature optimal complexity, beyond the time complexity metric.