In this paper, we investigate tensor recovery problems within the tensor singular value decomposition (t-SVD) framework. We propose the partial sum of the tubal nuclear norm (PSTNN) of a tensor. The PSTNN is a surrogate of the tensor tubal multi-rank. We build two PSTNN-based minimization models for two typical tensor recovery problems, i.e., the tensor completion and the tensor principal component analysis. We give two algorithms based on the alternating direction method of multipliers (ADMM) to solve proposed PSTNN-based tensor recovery models. Experimental results on the synthetic data and real-world data reveal the superior of the proposed PSTNN. •We propose the partial sum of the tubal nuclear norm (PSTNN) for tensor recovery.•The partial singular value thresholding (PSVT) is extended for complex matrices.•Two PSTNN minimization models are developed for TC and TRPCA problems.•Two efficient ADMM algorithms have been designed to solve the proposed models.•Extensive experiments are conducted on simulated and real-world data.