What's this engine fitted to? Assuming a riding mower.....
Is there any sound at all when the key is turned to 'START'? If you can hear a 'click' then the starter solenoid is being triggered, which eliminates safty switches and associated wiring.
Test the starter next - take a car jumper cable and connect the starter terminal to the battery+'ve terminal. If engine does NOT crank, then starter is bad. If it does, (and the 'click' test was OK), solenoid is bad.
If there was no click, check the solenoid by taking a wire from the small 'trigger' terminal on the solenoid to battery +'ve. If engine now cranks, you have a bad safty switch, ignition switch, or wiring.
If the equipment is a walk behind that you can't access all the relevant terminals on, let me know and I'll talk you through some alternative test procedures.