This PhD course introduces inverse problems governed by PDEs through a sequence of simple but revealing 1D case studies. We use elliptic and parabolic prototypes to understand what can (and cannot) be reconstructed from indirect measurements, highlighting three recurring themes: non-identifiability from boundary observations, nonlinearity of parameter-to-data maps, and instability mechanisms caused by operations such as differentiation and division in explicit reconstruction formulas. The backward heat equation serves as a paradigmatic example of severe ill-posedness, where high-frequency components are exponentially amplified, and it motivates stabilization strategies based on a priori information and regularization (e.g., spectral truncation), leading to conditional stability bounds and a clear resolution–stability trade-off. In the later part of the course we place these examples into a broader operator-theoretic framework, introducing compact operators, spectral theory, the singular value decomposition, and how these tools explain the structure and degree of ill-posedness in inverse problems.