This thesis aims to explore electromagnetic metasurfaces, focusing on their numerical and physical design in the microwave regime. The metasurfaces meticulously manipulate electromagnetic fields by synthesizing a set of auxiliary surface waves in the invisible region. Initially, a large plane-wave-to-surface-wave coupler is numerically and physically designed in three types of physical structures under an assumption of impenetrable surface. Moving beyond the assumption, this thesis presents a detailed theoretical analysis and design methodology for a penetrable metasurface. The scalar and tensorial penetrable metasurfaces have been developed for two applications: invisibility cloaking and leaky-wave antennas. First, thin and passive single-layer metasurfaces for invisibility cloaking are presented. The metasurfaces convert an incident plane wave into surface waves on the lit side and continuously leak the power as a plane wave with the aligned wavefront to the incident wave on the shadow side. The 2-D cylindrical cloaking techniques are demonstrated and experimentally validated for single- and dual-polarization. Extending the ideas, a 3-D spherical cloaking strategy has been developed with a physical design consisting of a crossed dipole array on a conducting dielectric shell. For leaky-wave antennas, this thesis introduces planar and conformal apertures. In the planar leaky-wave antenna designs, the design methodology achieves an accurate radiating performance by taking into account the constituent material losses which significantly have an impact on radiating performances in practice. Based on the design method, a completely flush planar leaky-wave antenna has been introduced as well. Designing leaky-wave antennas on a conformal surface is accomplished via forward field projection and total field synthesis method. Furthermore, a circularly-polarized leaky-wave antenna on a conformal surface is designed, demonstrating the scalability of the design approach.