There has been a growing interest in the commercialization of millimeter wave (mmW) technology as a part of the Fifth-Generation New Radio (5G-NR) wireless standardization efforts. In this direction, many sets of independent measurement campaigns show that wireless propagation at mmW carrier frequencies is only marginally worse than propagation at sub-6 GHz carrier frequencies for small-cell coverage --- one of the most important use-cases for 5G-NR. On the other hand, the biggest determinants of viability of mmW systems in practice are penetration and blockage of mmW signals through different materials in the scattering environment. With this background, the focus of this paper is on understanding the impact of blockage of mmW signals and reduced spatial coverage due to penetration through the human hand, body, vehicles, etc. Leveraging measurements with a 28 GHz mmW experimental prototype and electromagnetic simulation studies, we first propose statistical blockage models to capture the impact of the hand, human body and vehicles. We then study the time-scales at which mmW signals are disrupted by blockage (hand and human body). Our results show that these events can be attributed to physical movements and the time-scales corresponding to blockage are hence on the order of a few 100 ms or more. Building on this fundamental understanding, we finally consider the broader question of robustness of mmW beamforming to handle blockage. Network densification, subarray switching in a user equipment (UE) designed with multiple subarrays, fall back mechanisms such as codebook enhancements and switching to legacy carriers in non-standalone deployments, etc. can address blockage before it leads to a deleterious impact on the mmW link margin.