The task of Stance Detection involves discerning the stance expressed in a text towards a specific subject or target. Prior works have relied on existing transformer models that lack the capability to prioritize targets effectively. Consequently, these models yield similar performance regardless of whether we utilize or disregard target information, undermining the task's significance. To address this challenge, we introduce Stanceformer, a target-aware transformer model that incorporates enhanced attention towards the targets during both training and inference. Specifically, we design a \textit{Target Awareness} matrix that increases the self-attention scores assigned to the targets. We demonstrate the efficacy of the Stanceformer with various BERT-based models, including state-of-the-art models and Large Language Models (LLMs), and evaluate its performance across three stance detection datasets, alongside a zero-shot dataset. Our approach Stanceformer not only provides superior performance but also generalizes even to other domains, such as Aspect-based Sentiment Analysis. We make the code publicly available.\footnote{\scriptsize\url{https://github.com/kgarg8/Stanceformer}}