We present in this PhD thesis work various applications of digital holographic microscopy (DHM), an imaging technique based on coherent illumination which enables the recovery of the full complex wavefront, i.e. the amplitude and phase of a wave field which interacted with a specimen. The possibility to retrieve the phase information with DHM allows to measure surfaces with nanometric accuracy, or to employ it as an endogenous quantitative signal to assess the morphology of biological specimens. The technique has been developed during the past fifteen years to reach nowadays a mature state, where it can be used routinely for metrology applications for example. We study in this work advanced applications by taking advantage of this technique, while focusing on a specific measurement method of DHM, namely the off-axis configuration, which makes it possible to measure the complex wave field with one-shot capability through spatial encoding, thus enabling real-time detection. In a first part, we develop mathematical methods based on the fundamental model of holographic recording to suppress the so-called zero-order, which consists in intensity terms that coherent detection must suppress for complex wave retrieval. In the particular case of off-axis holography, the zero-order terms usually limit the spatial resolution because of the spatial encoding of the coherent signal. We first develop an iterative method which uses the fundamental relations between coherent and incoherent detection, in order to gradually suppress the zero order terms. In a second stage, we develop a non-iterative filtering method, based on nonlinear operators. The technique is based on the transfer to another filtering space through the use of the logarithm, and enables intrinsic suppression of the zero-order terms. Both methods present the advantage of not relying on any approximation, and are thus general for any off-axis holographic configuration. We show their applicability on various hologram types, and demonstrate that in the context of microscopy, their use can increase the spatial resolution of holography, in order to reach diffraction-limited imaging for any magnification. In a second part, we study potential applications of three-dimensional imaging through coherent detection by employing multiple acquisitions with a new scanning method. The coupling of tomographic reconstruction and quantitative phase imaging showed great potential in various published works, yielding to quantitative 3D refractive index distribution measured within biological specimens, and super-resolution imaging through synthetic aperture formalism. These methods are however still subjects to many issues, in particular due to practical limitations such as mechanical imprecision in the measurement protocols and the availability of flexible reconstruction algorithms. We study a new data acquisition method which eliminates the necessity of any scanning of the illumination pattern or object rotation during the acquisition, providing potentially a more stable acquisition protocol. We present results proving the principle of our approach by measuring the 3D refractive index distribution of pollen grain. In a third part, we applied DHM to the analysis of cell morphology and dynamics, applied in particular to neuronal cells. We couple the phase measurement with widely assessed methods such as dye probing or quantitative wide field fluorescence, in order to derive relevant biological indicators from DHM. Through the interpretation of the phase as an indicator of cell volume regulation, we derive criteria for early label-free cell death detection, where we show that cell monitoring with DHM makes it possible to detect cell non-viability at early stage by measuring deregulation mechanisms. We compare our methods with dyes for cell viability assessment, showing that DHM can detect cell death typically hours before usual dye probing procedures. We also couple the phase signal with intracellular ionic concentration imaging through fluorescence, showing that the phase measured on neuron cultures is intimately linked with ionic homeostasis and in particular transmembrane water movements accompanying ionic currents such as Ca2+ or Na+. We derive typical phase signatures related to the well-known Ca2+ bursts occurring during action potentials in neurons through stimulation with glutamate, one of the major neurotransmitters in the central nervous system.