runko is a new open-source plasma simulation framework implemented in C++ and Python. It is designed to function as an easy-to-extend general toolbox for simulating astrophysical plasmas with different theoretical and numerical models. Computationally intensive low-level kernels are written in modern C++ taking advantage of polymorphic classes, multiple inheritance, and template metaprogramming. High-level functionality is operated with Python scripts. The hybrid program design ensures good code performance together with ease of use. The framework has a modular object-oriented design that allows the user to easily add new numerical algorithms to the system. The code can be run on various computing platforms ranging from laptops (shared-memory systems) to massively parallel supercomputer architectures (distributed-memory systems). The framework supports heterogeneous multiphysics simulations in which different physical solvers can be combined and run simultaneously. Here, we showcase the framework's relativistic particle-in-cell (PIC) module by presenting (i) 1D simulations of relativistic Weibel instability, (ii) 2D simulations of relativistic kinetic turbulence in a suddenly stirred magnetically-dominated pair plasma, and (iii) 3D simulations of collisionless shocks in an unmagnetized medium.