This work introduces the BoundMPC strategy, an innovative online model-predictive path-following approach for robot manipulators. This joint-space trajectory planner allows the following of Cartesian reference paths in the end-effector's position and orientation, including via-points, within the desired asymmetric bounds of the orthogonal path error. These bounds encode the obstacle-free space and additional task-specific constraints in Cartesian space. Contrary to traditional path-following concepts, BoundMPC purposefully deviates from the Cartesian reference path in position and orientation to account for the robot's kinematics, leading to more successful task executions for Cartesian reference paths. Furthermore the simple reference path formulation is computationally efficient and allows for replanning during the robot's motion. This feature makes it possible to use this planner for dynamically changing environments and varying goals. The flexibility and performance of BoundMPC are experimentally demonstrated by five scenarios on a 7-DoF Kuka LBR iiwa 14 R820 robot. The first scenario shows the transfer of a larger object from a start to a goal pose through a confined space where the object must be tilted. The second scenario deals with grasping an object from a table where the grasping point changes during the robot's motion, and collisions with other obstacles in the scene must be avoided. The adaptability of BoundMPC is showcased in scenarios such as the opening of a drawer, the transfer of an open container, and the wiping of a table, where it effectively handles task-specific constraints. The last scenario highlights the possibility of accounting for collisions with the entire robot's kinematic chain. The code is readily available at https://github.com/thieso/boundmpc, inspiring you to explore its potential and adapt it to your specific robotic tasks.