Delay is known to be a critical performance metric for various real-world routing applications including multimedia communication and freight delivery. Provisioning delay-minimal (or at least delay-bounded) routing services for all traffic of an application is highly important. As a basic paradigm of networking, multi-path routing has been proven to be able to obtain lower delay performance than the single-path routing, since traffic congestions can be avoided. However, to our best knowledge, (i) many of existing delay-aware multi-path routing studies only consider the aggregate traffic delay. Considering that even the solution achieving the optimal aggregate traffic delay has a possibly unbounded delay performance for certain individual traffic unit, those studies may be insufficient in practice; besides, (ii) most existing studies which optimize or bound delays of all traffic are best-effort, where the achieved solutions have no theoretical performance guarantee. In this dissertation, we study four delay-aware multi-path routing problems, with the delay performances of all traffic taken into account. Three of them are in communication and one of them is in transportation. Note that our study differ from all related ones as we are the first to study the four fundamental problems to our best knowledge. Although we prove that our studied problems are all NP-hard, we design approximation algorithms with theoretical performance guarantee for solving each of them. To be specific, we claim the following contributions. Minimize maximum delay and average delay. First, we consider a single-unicast setting where in a multi-hop network a sender requires to use multiple paths to stream a flow at a fixed rate to a receiver. Two important delay metrics are the average sender-to-receiver delay and the maximum sender-to-receiver delay. Existing results say that the two delay metrics of a flow cannot be both within bounded-ratio gaps to the optimal. In comparison, we design three different flow solutions, each of which can minimize the two delay metrics simultaneously within a $(1/epsilon)$-ratio gap to the optimal, at a cost of only delivering $(1-epsilon)$-fraction of the flow, for any user-defined $epsilonin(0,1)$. The gap $(1/epsilon)$ is proven to be at least near-tight, and we further show that our solutions can be extended to the multiple-unicast setting. Minimize Age-of-Information (AoI). Second, we consider a single-unicast setting where in a multi-hop network a sender requires to use multiple paths to periodically send a batch of data to a receiver. We study a newly proposed delay-sensitive networking performance metric, AoI, defined as the elapsed time since the generation of the last received data. We consider the problem of minimizing AoI subject to throughput requirements, which we prove is NP-hard. We note that our AoI problem differs from existing ones in that we are the first to consider the batch generation of data and multi-path communication. We develop both an optimal algorithm with a pseudo-polynomial time complexity and an approximation framework with a polynomial time complexity. Our framework can build upon any polynomial-time $alpha$-approximation algorithm of the maximum delay minimization problem, to construct an $(alpha+c)$-approximate solution for minimizing AoI. Here $c$ is a constant dependent on throughput requirements. Maximize network utility. Third, we consider a multiple-unicast setting where in a multi-hop network there exist many network users. Each user requires a sender to use multiple paths to stream a flow to a receiver, incurring an utility that is a function of the experienced maximum delay or the achieved throughput. Our objective is to maximize the aggregate utility of all users under throughput requirements and maximum delay constraints. We observe that it is NP-complete either to construct an optimal solution under relaxed maximum delay constraints or relaxed throughput requirements, or to figure out a feasible solution with all constraints satisfied. Hence it is non-trivial even to obtain approximate solutions satisfying relaxed constraints in a polynomial time. We develop a polynomial-time approximation algorithm. Our algorithm obtains solutions with constant approximation ratios under realistic conditions, at the cost of violating constraints by up to constant-ratios. Minimize fuel consumption for a heavy truck to timely fulfill multiple transportation tasks. Finally, we consider a common truck operation scenario where a truck is driving in a national highway network to fulfill multiple transportation tasks in order. We study an NP-hard timely eco-routing problem of minimizing total fuel consumption under task pickup and delivery time window constraints. We note that optimizing task execution times is a new challenging design space for saving fuel in our multi-task setting, and it differentiates our study from existing ones under the single-task setting. We design a fast and efficient heuristic. We characterize conditions under which the solution of our heuristic must be optimal, and further prove its optimality gap in case the conditions are not met. We simulate a heavy-duty truck driving across the US national highway system, and empirically observe that the fuel consumption achieved by our heuristic can be $22%$ less than that achieved by the fastest-/shortest- path baselines. Furthermore, the fuel saving of our heuristic as compared to the baselines is robust to the number of tasks. / Doctor of Philosophy / We consider a network modeled as a directed graph, where it takes time for data to traverse each link in the network. It models many critical applications both in the communication area and in the transportation field. For example, both the European education network and the US national highway network can be modeled as directed graphs. We consider a scenario where a source node is required to send multiple (a set of) data packets to a destination node through the network as fast as possible, possibly using multiple source-to-destination paths. In this dissertation we study four problems all of which try to figure out routing solutions to send the set of data packets, with an objective of minimizing experienced travel time or subject to travel time constraints. Although all of our four problems are NP-hard, we design approximation algorithms to solve them and obtain solutions with theoretically bounded gaps as compared to the optimal. The first three problems are in the communication area, and the last problem is in the transportation field. We claim the following specific contributions. Minimize maximum delay and average delay. First, we consider the setting of simultaneously minimizing the average travel time and the worst (largest) travel time of sending the set of data packets from source to destination. Existing results say that the two metrics of travel time cannot be minimized to be both within bounded-ratio gaps to the optimal. As a comparison, we design three different routing solutions, each of which can minimize the two metrics of travel time simultaneously within a constant bounded ratio-gap to the optimal, but at a cost of only delivering a portion of the data. Minimize Age-of-Information (AoI). Second, we consider the problem of minimizing a newly proposed travel-time-sensitive performance metric, i.e., AoI, which is the elapsed time since the generation of the last received data. Our AoI study differs from existing ones in that we are the first to consider a set of data and multi-path routing. We develop both an optimal algorithm with a pseudo-polynomial time complexity and an approximation framework with a polynomial time complexity. Maximize network utility. Third, we consider a more general setting with multiple source destination pairs. Each source incurs a utility that is a function of the experienced travel time or the achieved throughput to send data to its destination. Our objective is to maximize the aggregate utility under throughput requirements and travel time constraints. We develop a polynomial-time approximation algorithm, at the cost of violating constraints by up to constant-ratios. It is non-trivial to design such algorithms, as we prove that it is NPcomplete either to construct an optimal solution under relaxed delay constraints or relaxed throughput requirements, or to figure out a feasible solution with all constraints satisfied. Minimize fuel consumption for a heavy truck to timely fulfill multiple transportation tasks. Finally, we consider a truck and multiple transportation tasks in order, where each task requires the truck to pick up cargoes at a source timely, and deliver them to a destination timely. The need of coordinating task execution times is a new challenging design space for saving fuel in our multi-task setting, and it differentiates our study from existing ones under the single-task setting. We design an efficient heuristic. We characterize conditions under which the solution of our heuristic must be optimal, and further prove its performance gap as compared to the optimal in case the conditions are not met.