The single most authoritative review is the 2016 consensus paper in Psychological Science in the Public Interest (Simons et al.), which evaluated over 130 studies. The conclusion: brain training works for the trained tasks, shows some near-transfer (to similar tasks), and limited far-transfer (to general intelligence or daily life). Marketing claims that training in one game raises IQ broadly are not supported.
However, the ACTIVE trial - a large NIH-funded randomized study following over 2,800 older adults for ten years (Rebok et al., Journal of the American Geriatrics Society, 2014) - found that targeted cognitive training produced durable improvements in reasoning and processing speed, with reduced risk of functional decline a decade later. This is a meaningful real-world benefit.
Working memory training (e.g., n-back tasks) shows the cleanest short-term gains; speed-of-processing training has the strongest long-term outcome data; and varied, challenging cognitive engagement of any kind is consistently associated with better cognitive aging (the 'cognitive reserve' hypothesis - Yaakov Stern, Columbia University).
Bottom line for everyday players: brain games are a low-cost, low-risk way to add daily cognitive engagement. Combine them with sleep, exercise, social connection, and learning anything new - that combination is where the evidence is strongest.