Peptides have shown increasing advantages and significant clinical value in drug discovery and development. With the development of high-throughput technologies and artificial intelligence (AI), machine learning (ML) methods for discovering new lead peptides have been expanded and incorporated into rational drug design. Predictions of peptide-protein interactions (PepPIs) and protein-protein interactions (PPIs) are both opportunities and challenges in computational biology, which will help to better understand the mechanisms of disease and provide the impetus for the discovery of lead peptides. This paper comprehensively reviews computational models for PepPI and PPI predictions. It begins with an introduction of various databases of peptide ligands and target proteins. Then it discusses data formats and feature representations for proteins and peptides. Furthermore, classical ML methods and emerging deep learning (DL) methods that can be used to train prediction models of PepPI and PPI are classified into four categories, and their advantages and disadvantages are analyzed. To assess the relative performance of different models, different validation protocols and evaluation indexes are discussed. The goal of this review is to help researchers quickly get started to develop computational frameworks using these integrated resources and eventually promote the discovery of lead peptides.