In this work, we propose a simultaneous localization and mapping (SLAM) system using a monocular camera and Ultra-wideband (UWB) sensors. Our system, referred to as VRSLAM, is a multi-stage framework that leverages the strengths and compensates for the weaknesses of each sensor. Firstly, we introduce a UWB-aided 7 degree-of-freedom (scale factor, 3D position, and 3D orientation) global alignment module to initialize the visual odometry (VO) system in the world frame defined by the UWB anchors. This module loosely fuses up-to-scale VO and ranging data using either a quadratically constrained quadratic programming (QCQP) or nonlinear least squares (NLS) algorithm based on whether a good initial guess is available. Secondly, we provide an accompanied theoretical analysis that includes the derivation and interpretation of the Fisher Information Matrix (FIM) and its determinant. Thirdly, we present UWBaided bundle adjustment (UBA) and UWB-aided pose graph optimization (UPGO) modules to improve short-term odometry accuracy, reduce long-term drift as well as correct any alignment and scale errors. Extensive simulations and experiments show that our solution outperforms UWB/camera-only and previous approaches, can quickly recover from tracking failure without relying on visual relocalization, and can effortlessly obtain a global map even if there are no loop closures.