Explore the foundations of the Selberg sieve in this comprehensive lecture from the Workshop on Additive Combinatorics. Delve into heuristics about twin primes, including naive approaches and the Hardy-Littlewood method. Examine probability concepts and learn about the goals for upcoming lectures. Investigate the simple Sieve of Eratosthenes and its extension to the Bateman-Horn Conjecture. Discover techniques for detecting primes in sequences and analyze examples. Study simple upper bound sieves and their application to twin primes. Conclude with an exploration of the Chinese remainder theorem and its implications for multiplicative functions in sieve theory.