Prompt engineering has become an important approach to making use of Large Language Models (LLMs) particularly for text and image generation tasks. This paper provides detailed insights into the prompt engineering techniques with a special focus on the large language models and their application in the domains of text and image generation. The challenge of prompt engineering is highlighted in the study, and the importance of the tuning of the performance of an LLM to given tasks is also demonstrated. The more advanced levels of complexity associated with LLMs are then reviewed, incorporating an analysis of the structure and the negative aspects of coherence as well as context in the generated outputs. The study also elucidates how the prompt engineering methodologies could be fitted and fine-tuned to enable LLMs to generate better textual content within that context. Examples of the modifications to the process of creating images using prompt engineering methods are presented, and the possibilities and restrictions of the methods are discussed. This work highlights the development prospects of modern technologies in the field of creation of text and images based on large language models, as well as the investigation of existing literature on the topic of prompt engineering of large language model-based systems.