This paper proposes a novel multi-mode pinching-antenna systems (PASS) framework. Multiple data streams can be transmitted within a single waveguide through multiple guided modes, thus facilitating efficient multi-user communications through the mode-domain multiplexing. A physic model is derived, which reveals the mode-selective power radiation feature of pinching antennas (PAs). A two-mode PASS enabled two-user downlink communication system is investigated. Considering the mode selectivity of PA power radiation, a practical PA grouping scheme is proposed, where each PA group matches with one specific guided mode and mainly radiates its signal sequentially. Depending on whether the guided mode leaks power to unmatched PAs or not, the proposed PA grouping scheme operates in either the non-leakage or weak-leakage regime. Based on this, the baseband beamforming and PA locations are jointly optimized for sum rate maximization, subject to each user's minimum rate requirement. 1) A simple two-PA case in non-leakage regime is first considered. To solve the formulated problem, a channel orthogonality based solution is proposed. The channel orthogonality is ensured by large-scale and wavelength-scale equality constraints on PA locations. Thus, the optimal beamforming reduces to maximum-ratio transmission (MRT). Moreover, the optimal PA locations are obtained via a Newton-based one-dimension search algorithm that enforces two-scale PA-location constraints by Newton's method. 2) A general multi-PA case in both non-leakage and weak-leakage regimes is further considered. A low-complexity particle-swarm optimization with zero-forcing beamforming (PSO-ZF) algorithm is developed, thus effectively tackling the high-oscillatory and strong-coupled problem. Simulation results demonstrate the superiority of the proposed multi-mode PASS over conventional single-mode PASS and fixed-antenna structures.