Petri Nets were introduced in the doctoral dissertation by K.A. Petri, titled "Kommunikation mit Automaten" and published in 1962 by University of Bonn. Petri Nets are graphical (the intuitive graphical modeling language) and mathematical (advanced formal analysis method) tool. The concurrence of performed actions is the natural phenomenon due to which Petri Nets are perceived as mathematical tool for modeling concurrent systems. The main idea of this theory was modified by many researchers according to their needs, owing to the unusual "flexibility" of this theory. The present monograph focuses on Petri Nets applications in two main areas: manufacturing (section 1) and computer science (section 2). These two areas have still huge influence on our lives and our world. The theory of Petri Nets is still developing: some directions of investigations are presented in section 3. And at the end there is section 4 including some infesting facts concerning application of Petri Nets in the public area: the analysis and control of public bicycle sharing systems. The monograph shows the results of research works performed with use of Petri Nets in science centers all over the world.