The automatic generation of stylized co-speech gestures has recently received increasing attention. Previous systems typically allow style control via predefined text labels or example motion clips, which are often not flexible enough to convey user intent accurately. In this work, we present GestureDiffuCLIP, a neural network framework for synthesizing realistic, stylized co-speech gestures with flexible style control. We leverage the power of the large-scale Contrastive-Language-Image-Pre-training (CLIP) model and present a novel CLIP-guided mechanism that extracts efficient style representations from multiple input modalities, such as a piece of text, an example motion clip, or a video. Our system learns a latent diffusion model to generate high-quality gestures and infuses the CLIP representations of style into the generator via an adaptive instance normalization (AdaIN) layer. We further devise a gesture-transcript alignment mechanism that ensures a semantically correct gesture generation based on contrastive learning. Our system can also be extended to allow fine-grained style control of individual body parts. We demonstrate an extensive set of examples showing the flexibility and generalizability of our model to a variety of style descriptions. In a user study, we show that our system outperforms the state-of-the-art approaches regarding human likeness, appropriateness, and style correctness.