Ultrasonic testing plays a crucial role in the fields of Non-Destructive Evaluation (NDE) and Structural Health Monitoring (SHM) by enabling the accurate detection and characterization of defects, material properties, and structural integrity without causing damage and, in many instances, without requiring disassembly. This dissertation presents recent advances in the science of ultrasonic testing for NDE and SHM, specifically in applications of (1) ultrasonic imaging of bulk solids and waveguide structures, and (2) identifying the elastic constants in composite panels by wave propagation inversion. Under the ultrasonic imaging topic, active modalities in the Synthetic Aperture Focusing Technique (SAFT) with Delay-And-Sum (DAS) algorithms are applied to image defects in bulk solids using a linear array mounted on a transducer wedge. Two prototype systems have been constructed to demonstrate quasi real-time 3D imaging of internal flaws in railroad rails using several enhancements in the SAFT methods for improved imaging quality and speed. Additionally, a passive SAFT scheme is proposed utilizing signal deconvolution with a Normalized Cross-Power Spectrum (NCPS) analysis in a dual-output system. This method enables efficient reconstruction of pure transfer functions (or Green’s functions) between receivers and demonstrates ultrafast imaging capabilities. Next, the Coherent White Noise Constraint (C-WNC) algorithm is introduced for super-resolution ultrasonic imaging in bulk solids using a linear array. Unlike traditional MUltiple SIgnal Classification (MUSIC) beamformers, C-WNC avoids subspace factorization while offering high dynamic range and precise focus on extended targets through its broadband and adaptive beamforming capabilities. The C-WNC algorithm is also applied to defect imaging in stiffened composites in aerospace structures with embedded sparse transducer array, exploiting the multimodal and dispersive nature of ultrasonic guided waves through data-driven match field modeling. Under the property identification topic, the dissertation presents a non-contact ultrasonic scanning system for the in-situ elastic constant inversion in stiffened composite aerospace panels subjected to impacts. This kind of non-destructive inspection that tracks the elastic constants is shown to be quite effective to detect and quantify impact damage in these structural components. Using the Semi-Analytical Finite Element (SAFE) method as a forward model, the inversion process minimizes the mismatch between experimental and predicted dispersion curves of the propagating guided waves via simulated-annealing optimization. A method is developed for efficient extraction of the dispersion curves by using only two measurement points, i.e. without the need for multiple measurement locations that are instead needed in conventional 2D-FFT techniques.