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As a result of recent advances in micro and nanotechnology, the tiny movements of nanoscale active and passive objects in fluids can be probed with ultrahigh sensitivity and time resolution. The overarching theme of this dissertation is to harness these movements in fluids in order to study fundamental fluid dynamics and develop novel biomedical devices. First, we use the oscillatory movements of nanocantilevers to investigate the scaling behavior of unsteady fluid flow. Here, our expansive experimental data and rigorous theoretical analysis suggest that a generalized scaling parameter combining the length and time scales of the flow governs the scaling. Second, we turn our attention to nanoscale movements of bacteria in a buffer. We develop a simple, robust and sensitive experimental method to detect and track the random movements of bacteria. Using this method, we show evidence that these random movements of bacteria correlate with their antibiotic susceptibility. In the first part of this thesis, we explore, through experimental and theoretical work, the breakdown of the Navier-Stokes equations in oscillatory fluid flows. The Navier-Stokes equations of hydrodynamics are based on two crucial assumptions. First, the fluid is approximated as a continuum, with a well-defined ``fluid particle." Second, the stress in the fluid is assumed to be a linear function of the rate-of-strain, resulting in a so-called Newtonian fluid. If a fluid such as an ideal gas is gradually rarefied, the Navier-Stokes equations begin to fail and a kinetic description of the flow becomes appropriate. The failure of the Navier-Stokes equations can be thought to take place via two different physical mechanisms: either the continuum hypothesis breaks down as a result of a finite size effect; or the local equilibrium is violated due to the high rate of strain. Our experimental approach is to create an unsteady flow by oscillating a finite-sized body in a gas and to measure the dissipation (or the drag force) acting on the body. By using micro and nanofabrication techniques, we independently tune the relevant linear dimensions and the frequencies of the oscillating bodies. We then measure the pressure-dependent dissipation of these micro/nano oscillators in three different gases, Helium, Nitrogen, and Argon. We observe that the scaling of the fluidic dissipation is governed by a subtle interplay between the length scale and the frequency, embodied respectively in the dimensionless Knudsen (Kn) and the Weissenberg ( Wi) numbers. We collapse all the experimental data using a single scaling parameter: Wi + Kn. This new dimensionless parameter, which can be regarded as a generalized Knudsen number, combines the relevant linear dimension and the frequency of the body; it is rooted in Galilean invariance and can be obtained rigorously from the Chapman-Enskog expansion of the Boltzmann equation. In the second part of the thesis, we turn to the movements of active micro-swimmers in a buffer. This portion of the work is motivated by a serious global public health problem: the rise of multi-drug resistant bacteria. One way to prevent this threat from growing is to treat bacterial infections with effective antibiotics using the minimum dosage. However, currently-used antibiotic susceptibility tests (ASTs), which determine whether or not bacterial isolates from a patient are susceptible to administered antibiotics, take too long. Here, we aim to develop a robust and rapid AST by exploiting a recently-observed microbiological phenomenon: various nanomechanical movements of bacteria subside promptly (within minutes) when the bacteria are exposed to an effective antibiotic. Our approach is to transduce bacterial movements into electrical voltage fluctuations in a microchannel filled with a buffer solution. When a small but constant current is driven through the microchannel, bacterial movements are converted into strong voltage fluctuations due to the fact that they modulate the effective microchannel diameter. Our experiments with E. coli show that the proposed detection method can provide antibiotic susceptibility results in ~1 hour, making it a promising rapid AST. Because this approach is based on a simple electrical measurement and does not require delicate process steps and instrumentation, it may eventually be used at the point of care.