A highly entertaining book on cellular automata:

A book I've enjoyed on Riemann surfaces:

Do the mathematical outputs of the human mind coincide with those of the ideal computer? An interesting Penrose venture: 

A singularity in the Euler equations:

A disproof of the strong cosmic censorship conjecture:

"You know, people think mathematics is complicated. Mathematics is the simple bit. It's the stuff we can understand. It's cats that are complicated." -- John Conway

(below: my cat)