During the past decade, using soft materials to build machines and robots has gained significant traction in the scientific domain. These soft devices target applications where human interaction, unstructured environments, and robust behavior are key. Despite these exciting developments, most of the current electronic control, intelligence, and power systems of soft robots are too bulky for embedded use and therefore limit their applicability. To enable bio-inspired forms of autonomy, we are in the process of developing soft machines embedded with smart fluidic (pneumatic) circuits that harness nonlinear mechanical and dynamical behavior to replace electronics and software. When such machines are designed properly, autonomy can emerge as a result of them interacting and responding to their environment. One of our recent studies involves an elastic valve with a slit that exhibits mechanical hysteretic behavior. We have demonstrated the ability to utilize this valve to convert a continuous flow into a pulsatile flow, a crucial requirement for enabling self-oscillating and programmable locomotion. We are currently in the process of exploiting such mechanofluidic instabilities to activate a soft robotic heart that we are developing within the Holland Hybrid Heart consortium. Similarly, alternative fluidic circuits feature nonlinear mechanical components such as kinking tubes, which exhibit instabilities that can be harnessed to enhance actuation frequencies and movement speed. From a behavioral point of view, these soft components change their behavior considerably in response to interactions with their environment, resulting in useful self-sensing behavior that can be utilized for dynamic synchronization, reprogrammable gaits, and potentially for autonomy.