A controversial method was Appel & Haken's use of computers in their 1977 proof of the Four Color Theorem that had bested Kempe, Tait, and generations of graph theorists. There's been a move to formal proof and, with improved computational power, memory, and techniques, results like Heule's determination in 2018 that 161 is the fifth Schur number. The Computer-assisted proof Wikipedia page is pretty good, and there's a new StackExchange site on this topic on the way.