The broadside radiation problem has plagued periodic leaky-wave antennas (LWAs) for more than four decades. A severe loss in performance, particularly in gain and efficiency, occurs when LWAs radiate in the broadside direction, which is the direction normal to the antenna plane. This broadside degradation has just recently been resolved in metamaterial LWAs, a special class of LWAs, as demonstrated by several research groups. This suggests that metamaterial LWAs exhibit extraordinary properties when compared with conventional LWAs. Intrigued by this finding, the author revisits conventional and metamaterial LWAs to answer the following research questions: why does the broadside problem exist, and how can the broadside problem be rigorously solved in periodic LWAs? These research questions are approached by studying symmetry and asymmetry in the LWA unit cells. The work unveils fundamental broadside radiation limitations and proposes a systematic resolution procedure. Beyond providing physical insight into the wave propagation mechanism of periodic LWAs, this thesis provides a practical framework for the design of periodic LWAs.