For many years Serge Lang has given talks to undergraduates on selected topics in mathematics which could be extracted at a level understandable by students who have had calculus. Written in a conversational tone, Lang now presents a collection of those talks: 1. Prime numbers, 2. The abc conjecture, 3. Global integration of locally integrable vector fields, 4. Approximation theorems of analysis, 5. Bruhat-Tits spaces, 6. Harmonic and symmetric polynomials.