This paper addresses the challenges encountered by grid-connected photovoltaic (PV) systems, including the stochastic behavior of the system, harmonic distortion, and variations in grid impedance. To this end, an in-depth technical and pedagogical analysis of three linear multivariable current control strategies is performed: proportional-integral (PI), proportional-resonant (PR), and deadbeat (DB). The study contributes to theoretical formulations, detailed system modeling, and controller tuning procedures, promoting a comprehensive understanding of their structures and performance. The strategies are investigated and compared in both the rotating ($dq$) and stationary ($\alpha \beta$) reference frames, offering a broad perspective on system behavior under various operating conditions. Additionally, an in-depth analysis of the PR controller is presented, highlighting its potential to regulate both positive- and negative-sequence components. This enables the development of more effective and robust tuning methodologies for steady-state and dynamic scenarios. The evaluation is conducted under three main conditions: steady-state operation, transient response to input power variations, and robustness analysis in the presence of grid parameter changes. The study examines the impact of each controller on the total harmonic distortion (THD) of the injected current, as well as on system stability margins and dynamic performance. Practical aspects that are often overlooked are also addressed, such as the modeling of the inverter and photovoltaic generator, the implementation of space vector pulse-width modulation (SVPWM), and the influence of the output LC filter capacitor. The control structures under analysis are validated through numerical simulations performed in MatLab^® software (R2021b) using dedicated computational routines, enabling the identification of strategies that enhance performance and ensure compliance of grid-connected photovoltaic systems.