The premise of the show is that former policeman Adrian Monk is an amazing detective but he has "problems" that keep him off of the force. Monk notices everything and can solve the most impossible crime while combating his personal demons.
At least that was the original premise. The show has drifted a lot in the last few years. It used to be about watching Monk solve the case and guessing what forgotten clue would be the murderer's undoing.
The new premise is to place Monk in new situations weekly. There are still crimes but they are secondary, more often an afterthought than the driving force behind the plot.
The current season opener is an example. Monk bought the house that had been a crime scene. The explanation for this was that Monk's psychiatrist died (as did the actor who played him) so he wanted to change everything in his life. In reality, this was just an excuse to place Monk in an uncomfortable situation.
Monk noticed that a ceiling light was off-center and had a handy-man move it. While doing this, the handy-man noticed several other problems and called in a helper. The two kept finding additional problems and tearing up the new home.
Monk's phobias include dirt, dust, and messiness so having two strangers tear up his spotless house was a nightmare for him. Eventually it turned out that they were looking for money that had been hidden in the house. Monk didn't do any actual detective work in this one.
This is either because the writers got lazy or because the management at USA wanted to stress Monk's character over the crimes. Either way, the show has jumped the shark.