Today's data networks are highly distributed and enormous in scale. The ability to measure them is vital to both network operators and end-users. Network measurement methods can broadly be classified into passive methods that rely on data collected at routers, and active methods based on observations of actively-injected probe packets. Active measurements, the focus of this dissertation, are attractive to end-users who, under the current network architecture, cannot access any measurement data collected at routers. Network operators use active measurements because they are easy to conduct, have low overhead and, in contrast to passive data collection methods, measure exactly what normal data packets experience. The most significant disadvantage of active measurements is the limited accuracy that has typically been achievable using them. One of the main reasons for this is in the need to be non-intrusive, thus leaving the measured systems uninfluenced by the observation, fundamentally affecting accuracy. In this dissertation, we use rigorous theoretical analysis to understand the impact of non-intrusiveness on active measurements and investigate how this theory translates into practice. Our investigation consists of three parts. In the first, we investigate sampling-related issues, i.e., when do we send probe packets and why? Our starting point is conventional wisdom that says that the "Poisson Arrivals See Time Averages (PASTA)" principle implies the need to use Poisson probing. We show that PASTA does not imply that Poisson probing is optimal because it ignores bias caused by probing intrusiveness and estimation variance. Using rigorous theory and simulations, we motivate rare probing, preferably at so-called mixing epochs, as a sound practical strategy. In the second part, we investigate if observed delays of (non-intrusive) probe pairs can be used to estimate cross-traffic properties in the single-hop case. Our starting point is the inability of prior works to estimate cross-traffic without hard-to-achieve timing control. We derive what can be estimated, in theory, and show that, under a well-motivated assumption, non-intrusive probe pairs can be used to estimate the entire distribution of cross-traffic in an intra-pair interval. Our third part is motivated by the apparent difficulty in designing non-intrusive active measurements robust to multi-hop queueing effects; We experience this first-hand with our single-hop cross-traffic estimators. We show that novel hop-dependent priority queueing primitives can be used to design Measurement-Friendly Networks (MFNs), networks in which accurate non-intrusive measurements, which are robust to multi-hop queueing effects, can be performed. Our primitives not only simplify network management tasks for network operators but are also easily deployable. In exploring MFNs, we find that nonpreemption and cross-traffic persistence cause unavoidable inaccuracies that represent, in a sense, fundamental limitations of active measurements.