The main goals of the tutorial are towards an understanding of the delay process in best-effort Internet for both non-congested and congested networks. A novel measurement system is reported for delay measurements in IP routers, which follows specifications of the IETF RFC 2679. The system is using both passive measurements and active probing and offers the possibility to measure and analyze different delay components of a router, e.g., packet processing delay, packet transmission time and queueing delay at the output link. Dedicated application-layer software is used to generate UDP traffic with TCP-like characteristics. Pareto traffic models are used to generate self-similar traffic in the link. The reported results are in form of several important statistics regarding processing and queueing delays of a router, router delay for a single data flow, router delay for more data flows as well as end-to-end delay for a chain of routers. We confirm results reported earlier about the fact that the delay in IP routers is generally influenced by traffic characteristics, link conditions and, at some extent, details in hardware implementation and different IOS releases. The delay in IP routers may also occasionally show extreme values, which are due to improper functioning of the routers. Furthermore, new results have been obtained that indicate that the delay in IP routers shows heavy-tailed characteristics, which can be well modeled with the help of three distributions, either in the form of single distribution or as a mixture of two distributions. There are several components contributing to the One-Way Transit Time (OWTT) in routers, i.e., processing delay, queueing delay and service time. Our results have shown that, e.g., the processing delay in a router can be modeled with the Normal or skewed Normal distribution, and the queueing delay is well modeled with a mixture of Normal distribution for the body probability mass and of Weibull distribution for the tail probability mass. It has been also observed that One-Way Transit Time (OWTT) is well modeled with the generalized Pareto distribution. Furthermore, OWTT has several component delays and it has been observed that the component delay distribution that is most dominant and heavy-tailed has a decisive influence on OWTT. To the best of our knowledge, this is the first time we understand the distributional properties of the delay process in an IP router.