Telecommunication networks are undergoing a disruptive transition toward distributed mobile edge networks with virtualized network functions (VNFs) [e.g., firewalls, intrusion detection systems (IDSs), and transcoders] within the proximity of users. This transition will enable network services, especially Internet-of-Things (IoT) applications, to be provisioned as network slices with sequences of VNFs, in order to guarantee the performance and security of their continuous data and control flows. In this article, we study the problems of delay-aware network slicing for multicasting traffic of IoT applications in edge networks. We first propose exact solutions by formulating the problems into integer linear programs (ILPs). We further devise an approximation algorithm with an approximation ratio for the problem of delay-aware network slicing for a single multicast slice, with the objective to minimize the implementation cost of the network slice subject to its delay requirement constraint. Given multiple multicast slicing requests, we also propose an efficient heuristic that admits as many user requests as possible, through exploring the impact of a nontrivial interplay of the total computing resource demand and delay requirements. We then investigate the problem of delay-oriented network slicing with given levels of delay guarantees, considering that different types of IoT applications have different levels of delay requirements, for which we propose an efficient heuristic based on reinforcement learning (RL). We finally evaluate the performance of the proposed algorithms through both simulations and implementations in a real testbed. The experimental results demonstrate that the proposed algorithms are promising.