This paper presents a visual SLAM system that uses both points and lines for robust camera localization, and simultaneously performs a piece-wise planar reconstruction (PPR) of the environment to provide a structural map in real-time. One of the biggest challenges in parallel tracking and mapping with a monocular camera is to keep the scale consistent when reconstructing the geometric primitives. This further introduces difficulties in graph optimization of the bundle adjustment (BA) step. We solve these problems by proposing several run-time optimizations on the reconstructed lines and planes. Our system is able to run with depth and stereo sensors in addition to the monocular setting. Our proposed SLAM tightly incorporates the semantic and geometric features to boost both frontend pose tracking and backend map optimization. We evaluate our system exhaustively on various datasets, and show that we outperform state-of-the-art methods in terms of trajectory precision. The code of PLP-SLAM has been made available in open-source for the research community (https://github.com/PeterFWS/Structure-PLP-SLAM).