The sixth-generation and beyond (B6G) networks are envisioned to support advanced applications that demand high-speed communication, high-precision sensing, and high-performance computing. To underpin this multi-functional evolution, energy- and cost-efficient programmable metasurfaces (PMs) have emerged as a promising technology for dynamically manipulating electromagnetic waves. This paper provides a comprehensive survey of representative multi-functional PM paradigms, with a specific focus on achieving \emph{full-space communication coverage}, \emph{ubiquitous sensing}, as well as \emph{intelligent signal processing and computing}. i) For simultaneously transmitting and reflecting surfaces (STARS)-enabled full-space communications, we elaborate on their operational protocols and pivotal applications in supporting efficient communications, physical layer security, unmanned aerial vehicle networks, and wireless power transfer. ii) For PM-underpinned ubiquitous sensing, we formulate the signal models for the PM-assisted architecture and systematically characterize its advantages in near-field and cooperative sensing, while transitioning to the PM-enabled transceiver architecture and demonstrating its superior performance in multi-band operations. iii) For advanced signal processing and computing, we explore the novel paradigm of stacked intelligent metasurfaces (SIMs), investigating their implementation in wave-domain analog processing and over-the-air mathematical computing. Finally, we identify key research challenges and envision future directions for multi-functional PMs towards B6G.