After several tens of minutes of the game, events will begin to occur. You can look up current events by clicking on the second icon on the bar in the upper-left corner of the screen. Most frequently, events concern trading of goods and they can quite considerably affect the price of goods in select cities. Also, events may concern warfare: you should then expect to encounter more army outposts along your path and searches for smuggling. Another example are epidemics: in cities they break out, people are willing to pay to be transported to another one. One of rare events are arrest warrants. While accepting a mission in the city hall, you will be rewarded for killing a bandit on a given route (often, he has a sizeable posse and you have to kill them all). You can then claim your bounty in any city hall and you will gain experience, and guns after the fight.
Occasionally, you can also encounter events that do not result from the current political or economic situation, but they concern your character. Often, you will be attacked by bandits - at which point you have to defend yourself (or pay to be left alone). If this happens, you switch into combat mode and control your crew. Apart from picking targets to destroy, you also have to ensure that your train is not set on fire and destroyed. As a result of that, you can lose your cargo, carriage or even a crew member.
Sometimes, you can try and escape. You then switch to the driver and you have to load coal into the burner, while minding turns, where you have to slow down (to prevent damage to the train) and not to blow up the boiler.
Brand new trains
Occasionally, you will also receive a message informing you about new locomotives and carriages being available in a given city. This offer is unlimited, but you have to remember what and where is on sale (if you are interested), because the messages are not saved anywhere.