The thesis focuses on the definition of bounds for maximum directivity and gain of antennas. The main goal is to establish an analytical formula for maximum super-directivity considering specific parameters like bandwidth and antenna size. The upper limit on directivity for self-resonant antennas within a minimum sphere is determined based on a given quality factor. The formulation, obtained through rigorous convex problem-solving, is expressed as a rapidly converging analytical series. Approximate closed-form formulas are derived, showing high accuracy in various ranges of the minimum circumscribed sphere's radius, including small and intermediate to large antennas. Special attention is given to small antennas, interpreting the solution as a combination of dipolar and quadrupolar Huygen's source contributions with closed-form coefficients. The solution maintains continuity to the maximum directivity between 3 and 8 while holding a constant Q. The challenge of achieving super-gain is addressed by assuming small losses in terms of surface resistance over the metalized surface of the minimum sphere circumscribing the antenna. The final closed-form formula indicates that maximum gain results from a summation similar to Harrington's sum for maximum directivity, with coefficients weighted by the radiation efficiency of each spherical harmonics. The formulation is extended to self-resonant antennas, providing a tighter bound for any losses. The thesis further explores the relationship between maximum directivity and Degrees of Freedom (DoF) of the fields.